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fdbfed3a2ac301ab22c709fbc63a5123ae3d14369e6fd7cb1018e881acbec562
|
def make_assembly_taxonomy_chart(tax_profile, genes, function_list, outfile, krona_path, metric='efpkg'):
"Writes XML file for taxonomy chart of assembly, one chart for all reads and separate charts\n for each function and generates Krona plot from it\n\n Args:\n tax_profile (:obj:TaxonomyProfile): taxonomy profile object\n genes (defaultdict[str,defaultdict[str,dict[str,float]]]): outer key is\n gene identifier, middle key is function identifier, inner key is in\n [metric, 'count', 'identity', 'coverage', 'Length', 'Completeness'],\n value is float (genes[gene_id][function_id][parameter_name] = parameter_value).\n function_list (list of str): function identifiers\n outfile (str): path for XML output\n krona_path (str): Krona Tools command\n metric (str): scoring metric (efpkg by default)\n "
with open(outfile, 'w') as out:
out.write('<krona key="false">\n')
out.write((('\t<attributes magnitude="' + metric) + '">\n'))
if (metric != 'readcount'):
out.write('\t\t<attribute display="Read count">readcount</attribute>\n')
out.write((((('\t\t<attribute display="Score:' + metric) + '">') + metric) + '</attribute>\n'))
out.write('\t\t<attribute display="Coverage" mono="true">coverage</attribute>\n')
out.write('\t\t<attribute display="Length" mono="true">Length</attribute>\n')
out.write(('\t\t<attribute display="CDS completeness %" mono="true">Completeness' + '</attribute>\n'))
out.write('\t\t<attribute display="Best hit identity %" mono="true">identity</attribute>\n')
out.write(('\t\t<attribute display="UniRef hit" hrefbase="https://www.uniprot.org/uniref/" ' + 'target="uniref" mono="true">best_hit</attribute>\n'))
out.write('\t</attributes>\n')
out.write(('\t<color attribute="identity" valueStart="50" valueEnd="100" hueStart="0" ' + 'hueEnd="240" default="true"></color>\n'))
out.write('\t<datasets>\n')
for function in function_list:
out.write((('\t\t<dataset>' + function) + '</dataset>\n'))
out.write('\t</datasets>\n')
offset = 1
out.write(get_assembly_tax_xml(tax_profile, genes, function_list, ROOT_TAXONOMY_ID, offset, metric))
out.write('</krona>')
html_file = (outfile + '.html')
krona_cmd = [krona_path, '-o', html_file, outfile]
run_external_program(krona_cmd)
|
Writes XML file for taxonomy chart of assembly, one chart for all reads and separate charts
for each function and generates Krona plot from it
Args:
tax_profile (:obj:TaxonomyProfile): taxonomy profile object
genes (defaultdict[str,defaultdict[str,dict[str,float]]]): outer key is
gene identifier, middle key is function identifier, inner key is in
[metric, 'count', 'identity', 'coverage', 'Length', 'Completeness'],
value is float (genes[gene_id][function_id][parameter_name] = parameter_value).
function_list (list of str): function identifiers
outfile (str): path for XML output
krona_path (str): Krona Tools command
metric (str): scoring metric (efpkg by default)
|
lib/fama/output/krona_xml_writer.py
|
make_assembly_taxonomy_chart
|
aekazakov/FamaProfiling
| 0 |
python
|
def make_assembly_taxonomy_chart(tax_profile, genes, function_list, outfile, krona_path, metric='efpkg'):
"Writes XML file for taxonomy chart of assembly, one chart for all reads and separate charts\n for each function and generates Krona plot from it\n\n Args:\n tax_profile (:obj:TaxonomyProfile): taxonomy profile object\n genes (defaultdict[str,defaultdict[str,dict[str,float]]]): outer key is\n gene identifier, middle key is function identifier, inner key is in\n [metric, 'count', 'identity', 'coverage', 'Length', 'Completeness'],\n value is float (genes[gene_id][function_id][parameter_name] = parameter_value).\n function_list (list of str): function identifiers\n outfile (str): path for XML output\n krona_path (str): Krona Tools command\n metric (str): scoring metric (efpkg by default)\n "
with open(outfile, 'w') as out:
out.write('<krona key="false">\n')
out.write((('\t<attributes magnitude="' + metric) + '">\n'))
if (metric != 'readcount'):
out.write('\t\t<attribute display="Read count">readcount</attribute>\n')
out.write((((('\t\t<attribute display="Score:' + metric) + '">') + metric) + '</attribute>\n'))
out.write('\t\t<attribute display="Coverage" mono="true">coverage</attribute>\n')
out.write('\t\t<attribute display="Length" mono="true">Length</attribute>\n')
out.write(('\t\t<attribute display="CDS completeness %" mono="true">Completeness' + '</attribute>\n'))
out.write('\t\t<attribute display="Best hit identity %" mono="true">identity</attribute>\n')
out.write(('\t\t<attribute display="UniRef hit" hrefbase="https://www.uniprot.org/uniref/" ' + 'target="uniref" mono="true">best_hit</attribute>\n'))
out.write('\t</attributes>\n')
out.write(('\t<color attribute="identity" valueStart="50" valueEnd="100" hueStart="0" ' + 'hueEnd="240" default="true"></color>\n'))
out.write('\t<datasets>\n')
for function in function_list:
out.write((('\t\t<dataset>' + function) + '</dataset>\n'))
out.write('\t</datasets>\n')
offset = 1
out.write(get_assembly_tax_xml(tax_profile, genes, function_list, ROOT_TAXONOMY_ID, offset, metric))
out.write('</krona>')
html_file = (outfile + '.html')
krona_cmd = [krona_path, '-o', html_file, outfile]
run_external_program(krona_cmd)
|
def make_assembly_taxonomy_chart(tax_profile, genes, function_list, outfile, krona_path, metric='efpkg'):
"Writes XML file for taxonomy chart of assembly, one chart for all reads and separate charts\n for each function and generates Krona plot from it\n\n Args:\n tax_profile (:obj:TaxonomyProfile): taxonomy profile object\n genes (defaultdict[str,defaultdict[str,dict[str,float]]]): outer key is\n gene identifier, middle key is function identifier, inner key is in\n [metric, 'count', 'identity', 'coverage', 'Length', 'Completeness'],\n value is float (genes[gene_id][function_id][parameter_name] = parameter_value).\n function_list (list of str): function identifiers\n outfile (str): path for XML output\n krona_path (str): Krona Tools command\n metric (str): scoring metric (efpkg by default)\n "
with open(outfile, 'w') as out:
out.write('<krona key="false">\n')
out.write((('\t<attributes magnitude="' + metric) + '">\n'))
if (metric != 'readcount'):
out.write('\t\t<attribute display="Read count">readcount</attribute>\n')
out.write((((('\t\t<attribute display="Score:' + metric) + '">') + metric) + '</attribute>\n'))
out.write('\t\t<attribute display="Coverage" mono="true">coverage</attribute>\n')
out.write('\t\t<attribute display="Length" mono="true">Length</attribute>\n')
out.write(('\t\t<attribute display="CDS completeness %" mono="true">Completeness' + '</attribute>\n'))
out.write('\t\t<attribute display="Best hit identity %" mono="true">identity</attribute>\n')
out.write(('\t\t<attribute display="UniRef hit" hrefbase="https://www.uniprot.org/uniref/" ' + 'target="uniref" mono="true">best_hit</attribute>\n'))
out.write('\t</attributes>\n')
out.write(('\t<color attribute="identity" valueStart="50" valueEnd="100" hueStart="0" ' + 'hueEnd="240" default="true"></color>\n'))
out.write('\t<datasets>\n')
for function in function_list:
out.write((('\t\t<dataset>' + function) + '</dataset>\n'))
out.write('\t</datasets>\n')
offset = 1
out.write(get_assembly_tax_xml(tax_profile, genes, function_list, ROOT_TAXONOMY_ID, offset, metric))
out.write('</krona>')
html_file = (outfile + '.html')
krona_cmd = [krona_path, '-o', html_file, outfile]
run_external_program(krona_cmd)<|docstring|>Writes XML file for taxonomy chart of assembly, one chart for all reads and separate charts
for each function and generates Krona plot from it
Args:
tax_profile (:obj:TaxonomyProfile): taxonomy profile object
genes (defaultdict[str,defaultdict[str,dict[str,float]]]): outer key is
gene identifier, middle key is function identifier, inner key is in
[metric, 'count', 'identity', 'coverage', 'Length', 'Completeness'],
value is float (genes[gene_id][function_id][parameter_name] = parameter_value).
function_list (list of str): function identifiers
outfile (str): path for XML output
krona_path (str): Krona Tools command
metric (str): scoring metric (efpkg by default)<|endoftext|>
|
78ae0002514fdcf8aa4278eb7c0838eed2f480eb180ccd38685493fb309627a7
|
def check_env_var(a_var, a_var_name):
' Check that an expected environment variable is actually present.\n\n :param a_var: Variable to be checked\n :param a_var_name: Name of environment variable that should be present\n :return: None; exit program if variable is not present\n '
if (a_var is None):
print(f'Environment variable {a_var_name} is not present!')
sys.exit(2)
|
Check that an expected environment variable is actually present.
:param a_var: Variable to be checked
:param a_var_name: Name of environment variable that should be present
:return: None; exit program if variable is not present
|
coll/mc_sim_coll_blog.py
|
check_env_var
|
aws-samples/optimize-your-monte-carlo-simulations-using-aws-batch
| 1 |
python
|
def check_env_var(a_var, a_var_name):
' Check that an expected environment variable is actually present.\n\n :param a_var: Variable to be checked\n :param a_var_name: Name of environment variable that should be present\n :return: None; exit program if variable is not present\n '
if (a_var is None):
print(f'Environment variable {a_var_name} is not present!')
sys.exit(2)
|
def check_env_var(a_var, a_var_name):
' Check that an expected environment variable is actually present.\n\n :param a_var: Variable to be checked\n :param a_var_name: Name of environment variable that should be present\n :return: None; exit program if variable is not present\n '
if (a_var is None):
print(f'Environment variable {a_var_name} is not present!')
sys.exit(2)<|docstring|>Check that an expected environment variable is actually present.
:param a_var: Variable to be checked
:param a_var_name: Name of environment variable that should be present
:return: None; exit program if variable is not present<|endoftext|>
|
71649988aaed8330217a9e22dcae38cc600746bcd002d117b5dc1d9f1bc5c272
|
def upload_file(file_name, bucket, object_name=None):
'Upload a file to an S3 bucket\n\n :param file_name: File to upload\n :param bucket: Bucket to upload to\n :param object_name: S3 object name. If not specified then file_name is used\n :return: True if file was uploaded, else False\n '
if (object_name is None):
object_name = os.path.basename(file_name)
s3_client = boto3.client('s3')
try:
response = s3_client.upload_file(file_name, bucket, object_name)
except ClientError as e:
logging.error(e)
return False
return True
|
Upload a file to an S3 bucket
:param file_name: File to upload
:param bucket: Bucket to upload to
:param object_name: S3 object name. If not specified then file_name is used
:return: True if file was uploaded, else False
|
coll/mc_sim_coll_blog.py
|
upload_file
|
aws-samples/optimize-your-monte-carlo-simulations-using-aws-batch
| 1 |
python
|
def upload_file(file_name, bucket, object_name=None):
'Upload a file to an S3 bucket\n\n :param file_name: File to upload\n :param bucket: Bucket to upload to\n :param object_name: S3 object name. If not specified then file_name is used\n :return: True if file was uploaded, else False\n '
if (object_name is None):
object_name = os.path.basename(file_name)
s3_client = boto3.client('s3')
try:
response = s3_client.upload_file(file_name, bucket, object_name)
except ClientError as e:
logging.error(e)
return False
return True
|
def upload_file(file_name, bucket, object_name=None):
'Upload a file to an S3 bucket\n\n :param file_name: File to upload\n :param bucket: Bucket to upload to\n :param object_name: S3 object name. If not specified then file_name is used\n :return: True if file was uploaded, else False\n '
if (object_name is None):
object_name = os.path.basename(file_name)
s3_client = boto3.client('s3')
try:
response = s3_client.upload_file(file_name, bucket, object_name)
except ClientError as e:
logging.error(e)
return False
return True<|docstring|>Upload a file to an S3 bucket
:param file_name: File to upload
:param bucket: Bucket to upload to
:param object_name: S3 object name. If not specified then file_name is used
:return: True if file was uploaded, else False<|endoftext|>
|
512a44fa9ddedd2e089856d28b04f208a3de525e2e1827c8dc9ce3976e737446
|
def get_input_csv(bucket_name, file_name):
' Download and read CSV file from an S3 bucket\n\n :param bucket_name: Bucket in which CSV file is located\n :param file_name: key name of the CSV file to read\n :return: DataFrame constructed from CSV file\n '
s3 = boto3.client('s3')
response = s3.get_object(Bucket=bucket_name, Key=file_name)
status = response.get('ResponseMetadata', {}).get('HTTPStatusCode')
if (status == 200):
print(f'Retrieved file {file_name} from bucket {bucket_name}')
return pd.read_csv(response.get('Body'), index_col=0)
else:
print(f'Error in retrieving file {file_name} from bucket {bucket_name}; {status}')
sys.exit(1)
|
Download and read CSV file from an S3 bucket
:param bucket_name: Bucket in which CSV file is located
:param file_name: key name of the CSV file to read
:return: DataFrame constructed from CSV file
|
coll/mc_sim_coll_blog.py
|
get_input_csv
|
aws-samples/optimize-your-monte-carlo-simulations-using-aws-batch
| 1 |
python
|
def get_input_csv(bucket_name, file_name):
' Download and read CSV file from an S3 bucket\n\n :param bucket_name: Bucket in which CSV file is located\n :param file_name: key name of the CSV file to read\n :return: DataFrame constructed from CSV file\n '
s3 = boto3.client('s3')
response = s3.get_object(Bucket=bucket_name, Key=file_name)
status = response.get('ResponseMetadata', {}).get('HTTPStatusCode')
if (status == 200):
print(f'Retrieved file {file_name} from bucket {bucket_name}')
return pd.read_csv(response.get('Body'), index_col=0)
else:
print(f'Error in retrieving file {file_name} from bucket {bucket_name}; {status}')
sys.exit(1)
|
def get_input_csv(bucket_name, file_name):
' Download and read CSV file from an S3 bucket\n\n :param bucket_name: Bucket in which CSV file is located\n :param file_name: key name of the CSV file to read\n :return: DataFrame constructed from CSV file\n '
s3 = boto3.client('s3')
response = s3.get_object(Bucket=bucket_name, Key=file_name)
status = response.get('ResponseMetadata', {}).get('HTTPStatusCode')
if (status == 200):
print(f'Retrieved file {file_name} from bucket {bucket_name}')
return pd.read_csv(response.get('Body'), index_col=0)
else:
print(f'Error in retrieving file {file_name} from bucket {bucket_name}; {status}')
sys.exit(1)<|docstring|>Download and read CSV file from an S3 bucket
:param bucket_name: Bucket in which CSV file is located
:param file_name: key name of the CSV file to read
:return: DataFrame constructed from CSV file<|endoftext|>
|
9696adca477e87492f3ccf9c29025870a80c2f39993f80332e2b622697c12bfe
|
def list_csv_files(bucket_name, folder_name=''):
' List all CSV files in a given folder in S3 bucket\n\n :param bucket_name: Bucket in which CSV files is located\n :param folder_name: Folder in bucket in which CSV files reside\n :return: List of names of CSV files in given bucket\n '
files = []
s3 = boto3.client('s3')
for f in s3.list_objects_v2(Bucket=bucket_name, Prefix=folder_name)['Contents']:
if (f['Key'].split('.')[(- 1)] == 'csv'):
files.append(f['Key'])
return files
|
List all CSV files in a given folder in S3 bucket
:param bucket_name: Bucket in which CSV files is located
:param folder_name: Folder in bucket in which CSV files reside
:return: List of names of CSV files in given bucket
|
coll/mc_sim_coll_blog.py
|
list_csv_files
|
aws-samples/optimize-your-monte-carlo-simulations-using-aws-batch
| 1 |
python
|
def list_csv_files(bucket_name, folder_name=):
' List all CSV files in a given folder in S3 bucket\n\n :param bucket_name: Bucket in which CSV files is located\n :param folder_name: Folder in bucket in which CSV files reside\n :return: List of names of CSV files in given bucket\n '
files = []
s3 = boto3.client('s3')
for f in s3.list_objects_v2(Bucket=bucket_name, Prefix=folder_name)['Contents']:
if (f['Key'].split('.')[(- 1)] == 'csv'):
files.append(f['Key'])
return files
|
def list_csv_files(bucket_name, folder_name=):
' List all CSV files in a given folder in S3 bucket\n\n :param bucket_name: Bucket in which CSV files is located\n :param folder_name: Folder in bucket in which CSV files reside\n :return: List of names of CSV files in given bucket\n '
files = []
s3 = boto3.client('s3')
for f in s3.list_objects_v2(Bucket=bucket_name, Prefix=folder_name)['Contents']:
if (f['Key'].split('.')[(- 1)] == 'csv'):
files.append(f['Key'])
return files<|docstring|>List all CSV files in a given folder in S3 bucket
:param bucket_name: Bucket in which CSV files is located
:param folder_name: Folder in bucket in which CSV files reside
:return: List of names of CSV files in given bucket<|endoftext|>
|
f09c5a4a22c789a381588bd91f766a639a56d0b23721d632a826c7429e98cb0e
|
def test_ir2fr():
'\n Test whether the frequency vector is correct.\n '
t = 1.0
fs = 100.0
f = 20.0
ts = np.arange(0, t, (1.0 / fs))
A = 5.0
x = (A * np.sin((((2.0 * np.pi) * f) * ts)))
(fv, fr) = ir2fr(x, fs)
assert_array_almost_equal(fv[np.abs(fr).argmax()], f)
assert_array_almost_equal(np.abs(fr).max(), A)
|
Test whether the frequency vector is correct.
|
tests/test_signal.py
|
test_ir2fr
|
andimarafioti/python-acoustics
| 1 |
python
|
def test_ir2fr():
'\n \n '
t = 1.0
fs = 100.0
f = 20.0
ts = np.arange(0, t, (1.0 / fs))
A = 5.0
x = (A * np.sin((((2.0 * np.pi) * f) * ts)))
(fv, fr) = ir2fr(x, fs)
assert_array_almost_equal(fv[np.abs(fr).argmax()], f)
assert_array_almost_equal(np.abs(fr).max(), A)
|
def test_ir2fr():
'\n \n '
t = 1.0
fs = 100.0
f = 20.0
ts = np.arange(0, t, (1.0 / fs))
A = 5.0
x = (A * np.sin((((2.0 * np.pi) * f) * ts)))
(fv, fr) = ir2fr(x, fs)
assert_array_almost_equal(fv[np.abs(fr).argmax()], f)
assert_array_almost_equal(np.abs(fr).max(), A)<|docstring|>Test whether the frequency vector is correct.<|endoftext|>
|
0429c965fd91e2a54d0f8bf9f53005518b3180a8aa7dcf861d44d16f61c4c36a
|
def test_LTI(self):
'\n Test whether it gives correct results for the LTI case.\n '
'Input signals.'
signals = [np.array([1, 2, 3, 4, 3, 2, 1], dtype='float'), np.array([1, 2, 3, 4, 3, 2, 1, 1], dtype='float')]
'Filters'
filters = [np.array([1, 2, 3, 4], dtype='float'), np.array([1, 2, 3, 4, 5], dtype='float')]
'Test for every combination of input signal and filter.'
for (u, h) in itertools.product(signals, filters):
H = np.tile(h, (len(u), 1)).T
np.testing.assert_array_almost_equal(convolveLTV(u, H), convolveLTI(u, h))
np.testing.assert_array_almost_equal(convolveLTV(u, H, mode='full'), convolveLTI(u, h, mode='full'))
np.testing.assert_array_almost_equal(convolveLTV(u, H, mode='valid'), convolveLTI(u, h, mode='valid'))
np.testing.assert_array_almost_equal(convolveLTV(u, H, mode='same'), convolveLTI(u, h, mode='same'))
|
Test whether it gives correct results for the LTI case.
|
tests/test_signal.py
|
test_LTI
|
andimarafioti/python-acoustics
| 1 |
python
|
def test_LTI(self):
'\n \n '
'Input signals.'
signals = [np.array([1, 2, 3, 4, 3, 2, 1], dtype='float'), np.array([1, 2, 3, 4, 3, 2, 1, 1], dtype='float')]
'Filters'
filters = [np.array([1, 2, 3, 4], dtype='float'), np.array([1, 2, 3, 4, 5], dtype='float')]
'Test for every combination of input signal and filter.'
for (u, h) in itertools.product(signals, filters):
H = np.tile(h, (len(u), 1)).T
np.testing.assert_array_almost_equal(convolveLTV(u, H), convolveLTI(u, h))
np.testing.assert_array_almost_equal(convolveLTV(u, H, mode='full'), convolveLTI(u, h, mode='full'))
np.testing.assert_array_almost_equal(convolveLTV(u, H, mode='valid'), convolveLTI(u, h, mode='valid'))
np.testing.assert_array_almost_equal(convolveLTV(u, H, mode='same'), convolveLTI(u, h, mode='same'))
|
def test_LTI(self):
'\n \n '
'Input signals.'
signals = [np.array([1, 2, 3, 4, 3, 2, 1], dtype='float'), np.array([1, 2, 3, 4, 3, 2, 1, 1], dtype='float')]
'Filters'
filters = [np.array([1, 2, 3, 4], dtype='float'), np.array([1, 2, 3, 4, 5], dtype='float')]
'Test for every combination of input signal and filter.'
for (u, h) in itertools.product(signals, filters):
H = np.tile(h, (len(u), 1)).T
np.testing.assert_array_almost_equal(convolveLTV(u, H), convolveLTI(u, h))
np.testing.assert_array_almost_equal(convolveLTV(u, H, mode='full'), convolveLTI(u, h, mode='full'))
np.testing.assert_array_almost_equal(convolveLTV(u, H, mode='valid'), convolveLTI(u, h, mode='valid'))
np.testing.assert_array_almost_equal(convolveLTV(u, H, mode='same'), convolveLTI(u, h, mode='same'))<|docstring|>Test whether it gives correct results for the LTI case.<|endoftext|>
|
b352d6cc6626731edff86dfe3803eff6cc38e687c8bbe040de46c3c781a8d6fe
|
def test_LTV(self):
'\n Test whether it gives correct results for the LTV case.\n '
'Input signal'
u = np.array([1, 1, 1])
'Impulse responses where each column represents an impulse response.'
C = np.array([[1, 0, 0], [2, 1, 1]])
'The result calculated manually.'
y_manual = np.array([1, 2, 1, 1])
y_ltv = convolveLTV(u, C)
np.testing.assert_array_equal(y_ltv, y_manual)
|
Test whether it gives correct results for the LTV case.
|
tests/test_signal.py
|
test_LTV
|
andimarafioti/python-acoustics
| 1 |
python
|
def test_LTV(self):
'\n \n '
'Input signal'
u = np.array([1, 1, 1])
'Impulse responses where each column represents an impulse response.'
C = np.array([[1, 0, 0], [2, 1, 1]])
'The result calculated manually.'
y_manual = np.array([1, 2, 1, 1])
y_ltv = convolveLTV(u, C)
np.testing.assert_array_equal(y_ltv, y_manual)
|
def test_LTV(self):
'\n \n '
'Input signal'
u = np.array([1, 1, 1])
'Impulse responses where each column represents an impulse response.'
C = np.array([[1, 0, 0], [2, 1, 1]])
'The result calculated manually.'
y_manual = np.array([1, 2, 1, 1])
y_ltv = convolveLTV(u, C)
np.testing.assert_array_equal(y_ltv, y_manual)<|docstring|>Test whether it gives correct results for the LTV case.<|endoftext|>
|
9aadc79c140f1c2de15ef59ad2763534764cbf09e8d50c1361a53dfd626e230a
|
def test_construction_1(self):
'Using center.'
x = np.arange(10.0, 20.0, 2.0)
b = EqualBand(x)
assert_array_equal(b.center, x)
|
Using center.
|
tests/test_signal.py
|
test_construction_1
|
andimarafioti/python-acoustics
| 1 |
python
|
def test_construction_1(self):
x = np.arange(10.0, 20.0, 2.0)
b = EqualBand(x)
assert_array_equal(b.center, x)
|
def test_construction_1(self):
x = np.arange(10.0, 20.0, 2.0)
b = EqualBand(x)
assert_array_equal(b.center, x)<|docstring|>Using center.<|endoftext|>
|
7cda07980a9931641085e642cd27d93af04c9fe95de794e689b02b4d4e756cd8
|
def test_construction_2(self):
'Using fstart, fstop and fbands'
x = np.arange(10.0, 20.0, 2.0)
fstart = x[0]
fstop = x[(- 1)]
nbands = len(x)
b = EqualBand(fstart=fstart, fstop=fstop, nbands=nbands)
assert_array_equal(b.center, x)
|
Using fstart, fstop and fbands
|
tests/test_signal.py
|
test_construction_2
|
andimarafioti/python-acoustics
| 1 |
python
|
def test_construction_2(self):
x = np.arange(10.0, 20.0, 2.0)
fstart = x[0]
fstop = x[(- 1)]
nbands = len(x)
b = EqualBand(fstart=fstart, fstop=fstop, nbands=nbands)
assert_array_equal(b.center, x)
|
def test_construction_2(self):
x = np.arange(10.0, 20.0, 2.0)
fstart = x[0]
fstop = x[(- 1)]
nbands = len(x)
b = EqualBand(fstart=fstart, fstop=fstop, nbands=nbands)
assert_array_equal(b.center, x)<|docstring|>Using fstart, fstop and fbands<|endoftext|>
|
3ae18da460af76c7321897f8ce75ad282df0b036e28949857f0a7b87bb558fd0
|
def test_construction_3(self):
'Using fstart, fstop and bandwidth'
x = np.arange(10.0, 20.0, 2.0)
fstart = x[0]
fstop = x[(- 1)]
bandwidth = np.diff(x)[0]
b = EqualBand(fstart=fstart, fstop=fstop, bandwidth=bandwidth)
assert_array_equal(b.center, x)
|
Using fstart, fstop and bandwidth
|
tests/test_signal.py
|
test_construction_3
|
andimarafioti/python-acoustics
| 1 |
python
|
def test_construction_3(self):
x = np.arange(10.0, 20.0, 2.0)
fstart = x[0]
fstop = x[(- 1)]
bandwidth = np.diff(x)[0]
b = EqualBand(fstart=fstart, fstop=fstop, bandwidth=bandwidth)
assert_array_equal(b.center, x)
|
def test_construction_3(self):
x = np.arange(10.0, 20.0, 2.0)
fstart = x[0]
fstop = x[(- 1)]
bandwidth = np.diff(x)[0]
b = EqualBand(fstart=fstart, fstop=fstop, bandwidth=bandwidth)
assert_array_equal(b.center, x)<|docstring|>Using fstart, fstop and bandwidth<|endoftext|>
|
e9e42bbf6125a0c027a9484245fd7e0873ac54d47a45da5447deb70a0a56f8cb
|
def write_history(line):
'ζζζ°ηεε²εε¨ζεζΉ'
line = line.strip()
if (line == ''):
return
with open(history_path, 'r+', encoding='utf8') as w:
content = w.read()
w.seek(0, 0)
w.write(((line + '\n') + content))
|
ζζζ°ηεε²εε¨ζεζΉ
|
main.py
|
write_history
|
Taokyla/MikanToAria2
| 5 |
python
|
def write_history(line):
line = line.strip()
if (line == ):
return
with open(history_path, 'r+', encoding='utf8') as w:
content = w.read()
w.seek(0, 0)
w.write(((line + '\n') + content))
|
def write_history(line):
line = line.strip()
if (line == ):
return
with open(history_path, 'r+', encoding='utf8') as w:
content = w.read()
w.seek(0, 0)
w.write(((line + '\n') + content))<|docstring|>ζζζ°ηεε²εε¨ζεζΉ<|endoftext|>
|
54fa0cec1c7b3cf3c90d11bdbda9cc7e34129bf638ad2e94c63f48943d02dec9
|
def apply_transforms(fixed, moving, transformlist, interpolator='linear', imagetype=0, whichtoinvert=None, compose=None, verbose=False, **kwargs):
"\n Apply a transform list to map an image from one domain to another. \n In image registration, one computes mappings between (usually) pairs \n of images. These transforms are often a sequence of increasingly \n complex maps, e.g. from translation, to rigid, to affine to deformation. \n The list of such transforms is passed to this function to interpolate one \n image domain into the next image domain, as below. The order matters \n strongly and the user is advised to familiarize with the standards \n established in examples.\n\n ANTsR function: `antsApplyTransforms`\n\n Arguments\n ---------\n fixed : ANTsImage\n fixed image defining domain into which the moving image is transformed.\n \n moving : AntsImage\n moving image to be mapped to fixed space.\n \n transformlist : list of strings \n list of transforms generated by ants.registration where each transform is a filename.\n \n interpolator : string \n Choice of interpolator. Supports partial matching.\n linear\n nearestNeighbor\n multiLabel for label images but genericlabel is preferred\n gaussian\n bSpline\n cosineWindowedSinc\n welchWindowedSinc\n hammingWindowedSinc\n lanczosWindowedSinc\n genericLabel use this for label images\n \n imagetype : integer \n choose 0/1/2/3 mapping to scalar/vector/tensor/time-series\n \n whichtoinvert : list of booleans (optional) \n Must be same length as transformlist.\n whichtoinvert[i] is True if transformlist[i] is a matrix, \n and the matrix should be inverted. If transformlist[i] is a \n warp field, whichtoinvert[i] must be False.\n If the transform list is a matrix followed by a warp field, \n whichtoinvert defaults to (True,False). Otherwise it defaults \n to [False]*len(transformlist)).\n \n compose : string (optional)\n if it is a string pointing to a valid file location, \n this will force the function to return a composite transformation filename.\n \n verbose : boolean\n print command and run verbose application of transform.\n \n kwargs : keyword arguments \n extra parameters\n \n Returns\n -------\n ANTsImage or string (transformation filename)\n\n Example\n -------\n >>> import ants\n >>> fixed = ants.image_read( ants.get_ants_data('r16') )\n >>> moving = ants.image_read( ants.get_ants_data('r64') )\n >>> fixed = ants.resample_image(fixed, (64,64), 1, 0)\n >>> moving = ants.resample_image(moving, (64,64), 1, 0)\n >>> mytx = ants.registration(fixed=fixed , moving=moving ,\n type_of_transform = 'SyN' )\n >>> mywarpedimage = ants.apply_transforms( fixed=fixed, moving=moving,\n transformlist=mytx['fwdtransforms'] )\n "
if ((not isinstance(transformlist, (tuple, list))) and (transformlist is not None)):
transformlist = [transformlist]
accepted_interpolators = {'linear', 'nearestNeighbor', 'multiLabel', 'gaussian', 'bSpline', 'cosineWindowedSinc', 'welchWindowedSinc', 'hammingWindowedSinc', 'lanczosWindowedSinc', 'genericLabel'}
if (interpolator not in accepted_interpolators):
raise ValueError(('interpolator not supported - see %s' % accepted_interpolators))
args = [fixed, moving, transformlist, interpolator]
if (not isinstance(fixed, str)):
if (isinstance(fixed, iio.ANTsImage) and isinstance(moving, iio.ANTsImage)):
for tl_path in transformlist:
if (not os.path.exists(tl_path)):
raise Exception(('Transform %s does not exist' % tl_path))
inpixeltype = fixed.pixeltype
fixed = fixed.clone('float')
moving = moving.clone('float')
warpedmovout = moving.clone()
f = fixed
m = moving
if ((moving.dimension == 4) and (fixed.dimension == 3) and (imagetype == 0)):
raise Exception('Set imagetype 3 to transform time series images.')
wmo = warpedmovout
mytx = []
if ((whichtoinvert is None) or (isinstance(whichtoinvert, (tuple, list)) and (sum([(w is not None) for w in whichtoinvert]) == 0))):
if ((len(transformlist) == 2) and ('.mat' in transformlist[0]) and ('.mat' not in transformlist[1])):
whichtoinvert = (True, False)
else:
whichtoinvert = tuple(([False] * len(transformlist)))
if (len(whichtoinvert) != len(transformlist)):
raise ValueError('Transform list and inversion list must be the same length')
for i in range(len(transformlist)):
ismat = False
if ('.mat' in transformlist[0]):
ismat = True
if (whichtoinvert[i] and (not ismat)):
raise ValueError(('Cannot invert transform %i (%s) because it is not a matrix' % (i, transformlist[i])))
if whichtoinvert[i]:
mytx = (mytx + ['-t', ('[%s,1]' % transformlist[i])])
else:
mytx = (mytx + ['-t', transformlist[i]])
if (compose is None):
args = ['-d', fixed.dimension, '-i', m, '-o', wmo, '-r', f, '-n', interpolator]
args = (args + mytx)
tfn = (('%scomptx.nii.gz' % compose) if (compose is not None) else 'NA')
if (compose is not None):
mycompo = ('[%s,1]' % tfn)
args = ['-d', fixed.dimension, '-i', m, '-o', mycompo, '-r', f, '-n', interpolator]
args = (args + mytx)
myargs = utils._int_antsProcessArguments(args)
for jj in range(len(myargs)):
if (myargs[jj] is not None):
if (myargs[jj] == '-'):
myargs2 = ([None] * (len(myargs) - 1))
myargs2[:(jj - 1)] = myargs[:(jj - 1)]
myargs2[jj:(len(myargs) - 1)] = myargs[(jj + 1):len(myargs)]
myargs = myargs2
myverb = int(verbose)
if verbose:
print(myargs)
processed_args = (myargs + ['-z', str(1), '-v', str(myverb), '--float', str(1), '-e', str(imagetype)])
libfn = utils.get_lib_fn('antsApplyTransforms')
libfn(processed_args)
if (compose is None):
return warpedmovout.clone(inpixeltype)
elif os.path.exists(tfn):
return tfn
else:
return None
else:
return 1
else:
args = (args + ['-z', 1, '--float', 1, '-e', imagetype])
processed_args = utils._int_antsProcessArguments(args)
libfn = utils.get_lib_fn('antsApplyTransforms')
libfn(processed_args)
|
Apply a transform list to map an image from one domain to another.
In image registration, one computes mappings between (usually) pairs
of images. These transforms are often a sequence of increasingly
complex maps, e.g. from translation, to rigid, to affine to deformation.
The list of such transforms is passed to this function to interpolate one
image domain into the next image domain, as below. The order matters
strongly and the user is advised to familiarize with the standards
established in examples.
ANTsR function: `antsApplyTransforms`
Arguments
---------
fixed : ANTsImage
fixed image defining domain into which the moving image is transformed.
moving : AntsImage
moving image to be mapped to fixed space.
transformlist : list of strings
list of transforms generated by ants.registration where each transform is a filename.
interpolator : string
Choice of interpolator. Supports partial matching.
linear
nearestNeighbor
multiLabel for label images but genericlabel is preferred
gaussian
bSpline
cosineWindowedSinc
welchWindowedSinc
hammingWindowedSinc
lanczosWindowedSinc
genericLabel use this for label images
imagetype : integer
choose 0/1/2/3 mapping to scalar/vector/tensor/time-series
whichtoinvert : list of booleans (optional)
Must be same length as transformlist.
whichtoinvert[i] is True if transformlist[i] is a matrix,
and the matrix should be inverted. If transformlist[i] is a
warp field, whichtoinvert[i] must be False.
If the transform list is a matrix followed by a warp field,
whichtoinvert defaults to (True,False). Otherwise it defaults
to [False]*len(transformlist)).
compose : string (optional)
if it is a string pointing to a valid file location,
this will force the function to return a composite transformation filename.
verbose : boolean
print command and run verbose application of transform.
kwargs : keyword arguments
extra parameters
Returns
-------
ANTsImage or string (transformation filename)
Example
-------
>>> import ants
>>> fixed = ants.image_read( ants.get_ants_data('r16') )
>>> moving = ants.image_read( ants.get_ants_data('r64') )
>>> fixed = ants.resample_image(fixed, (64,64), 1, 0)
>>> moving = ants.resample_image(moving, (64,64), 1, 0)
>>> mytx = ants.registration(fixed=fixed , moving=moving ,
type_of_transform = 'SyN' )
>>> mywarpedimage = ants.apply_transforms( fixed=fixed, moving=moving,
transformlist=mytx['fwdtransforms'] )
|
ants/registration/apply_transforms.py
|
apply_transforms
|
vascosa/ANTsPy
| 0 |
python
|
def apply_transforms(fixed, moving, transformlist, interpolator='linear', imagetype=0, whichtoinvert=None, compose=None, verbose=False, **kwargs):
"\n Apply a transform list to map an image from one domain to another. \n In image registration, one computes mappings between (usually) pairs \n of images. These transforms are often a sequence of increasingly \n complex maps, e.g. from translation, to rigid, to affine to deformation. \n The list of such transforms is passed to this function to interpolate one \n image domain into the next image domain, as below. The order matters \n strongly and the user is advised to familiarize with the standards \n established in examples.\n\n ANTsR function: `antsApplyTransforms`\n\n Arguments\n ---------\n fixed : ANTsImage\n fixed image defining domain into which the moving image is transformed.\n \n moving : AntsImage\n moving image to be mapped to fixed space.\n \n transformlist : list of strings \n list of transforms generated by ants.registration where each transform is a filename.\n \n interpolator : string \n Choice of interpolator. Supports partial matching.\n linear\n nearestNeighbor\n multiLabel for label images but genericlabel is preferred\n gaussian\n bSpline\n cosineWindowedSinc\n welchWindowedSinc\n hammingWindowedSinc\n lanczosWindowedSinc\n genericLabel use this for label images\n \n imagetype : integer \n choose 0/1/2/3 mapping to scalar/vector/tensor/time-series\n \n whichtoinvert : list of booleans (optional) \n Must be same length as transformlist.\n whichtoinvert[i] is True if transformlist[i] is a matrix, \n and the matrix should be inverted. If transformlist[i] is a \n warp field, whichtoinvert[i] must be False.\n If the transform list is a matrix followed by a warp field, \n whichtoinvert defaults to (True,False). Otherwise it defaults \n to [False]*len(transformlist)).\n \n compose : string (optional)\n if it is a string pointing to a valid file location, \n this will force the function to return a composite transformation filename.\n \n verbose : boolean\n print command and run verbose application of transform.\n \n kwargs : keyword arguments \n extra parameters\n \n Returns\n -------\n ANTsImage or string (transformation filename)\n\n Example\n -------\n >>> import ants\n >>> fixed = ants.image_read( ants.get_ants_data('r16') )\n >>> moving = ants.image_read( ants.get_ants_data('r64') )\n >>> fixed = ants.resample_image(fixed, (64,64), 1, 0)\n >>> moving = ants.resample_image(moving, (64,64), 1, 0)\n >>> mytx = ants.registration(fixed=fixed , moving=moving ,\n type_of_transform = 'SyN' )\n >>> mywarpedimage = ants.apply_transforms( fixed=fixed, moving=moving,\n transformlist=mytx['fwdtransforms'] )\n "
if ((not isinstance(transformlist, (tuple, list))) and (transformlist is not None)):
transformlist = [transformlist]
accepted_interpolators = {'linear', 'nearestNeighbor', 'multiLabel', 'gaussian', 'bSpline', 'cosineWindowedSinc', 'welchWindowedSinc', 'hammingWindowedSinc', 'lanczosWindowedSinc', 'genericLabel'}
if (interpolator not in accepted_interpolators):
raise ValueError(('interpolator not supported - see %s' % accepted_interpolators))
args = [fixed, moving, transformlist, interpolator]
if (not isinstance(fixed, str)):
if (isinstance(fixed, iio.ANTsImage) and isinstance(moving, iio.ANTsImage)):
for tl_path in transformlist:
if (not os.path.exists(tl_path)):
raise Exception(('Transform %s does not exist' % tl_path))
inpixeltype = fixed.pixeltype
fixed = fixed.clone('float')
moving = moving.clone('float')
warpedmovout = moving.clone()
f = fixed
m = moving
if ((moving.dimension == 4) and (fixed.dimension == 3) and (imagetype == 0)):
raise Exception('Set imagetype 3 to transform time series images.')
wmo = warpedmovout
mytx = []
if ((whichtoinvert is None) or (isinstance(whichtoinvert, (tuple, list)) and (sum([(w is not None) for w in whichtoinvert]) == 0))):
if ((len(transformlist) == 2) and ('.mat' in transformlist[0]) and ('.mat' not in transformlist[1])):
whichtoinvert = (True, False)
else:
whichtoinvert = tuple(([False] * len(transformlist)))
if (len(whichtoinvert) != len(transformlist)):
raise ValueError('Transform list and inversion list must be the same length')
for i in range(len(transformlist)):
ismat = False
if ('.mat' in transformlist[0]):
ismat = True
if (whichtoinvert[i] and (not ismat)):
raise ValueError(('Cannot invert transform %i (%s) because it is not a matrix' % (i, transformlist[i])))
if whichtoinvert[i]:
mytx = (mytx + ['-t', ('[%s,1]' % transformlist[i])])
else:
mytx = (mytx + ['-t', transformlist[i]])
if (compose is None):
args = ['-d', fixed.dimension, '-i', m, '-o', wmo, '-r', f, '-n', interpolator]
args = (args + mytx)
tfn = (('%scomptx.nii.gz' % compose) if (compose is not None) else 'NA')
if (compose is not None):
mycompo = ('[%s,1]' % tfn)
args = ['-d', fixed.dimension, '-i', m, '-o', mycompo, '-r', f, '-n', interpolator]
args = (args + mytx)
myargs = utils._int_antsProcessArguments(args)
for jj in range(len(myargs)):
if (myargs[jj] is not None):
if (myargs[jj] == '-'):
myargs2 = ([None] * (len(myargs) - 1))
myargs2[:(jj - 1)] = myargs[:(jj - 1)]
myargs2[jj:(len(myargs) - 1)] = myargs[(jj + 1):len(myargs)]
myargs = myargs2
myverb = int(verbose)
if verbose:
print(myargs)
processed_args = (myargs + ['-z', str(1), '-v', str(myverb), '--float', str(1), '-e', str(imagetype)])
libfn = utils.get_lib_fn('antsApplyTransforms')
libfn(processed_args)
if (compose is None):
return warpedmovout.clone(inpixeltype)
elif os.path.exists(tfn):
return tfn
else:
return None
else:
return 1
else:
args = (args + ['-z', 1, '--float', 1, '-e', imagetype])
processed_args = utils._int_antsProcessArguments(args)
libfn = utils.get_lib_fn('antsApplyTransforms')
libfn(processed_args)
|
def apply_transforms(fixed, moving, transformlist, interpolator='linear', imagetype=0, whichtoinvert=None, compose=None, verbose=False, **kwargs):
"\n Apply a transform list to map an image from one domain to another. \n In image registration, one computes mappings between (usually) pairs \n of images. These transforms are often a sequence of increasingly \n complex maps, e.g. from translation, to rigid, to affine to deformation. \n The list of such transforms is passed to this function to interpolate one \n image domain into the next image domain, as below. The order matters \n strongly and the user is advised to familiarize with the standards \n established in examples.\n\n ANTsR function: `antsApplyTransforms`\n\n Arguments\n ---------\n fixed : ANTsImage\n fixed image defining domain into which the moving image is transformed.\n \n moving : AntsImage\n moving image to be mapped to fixed space.\n \n transformlist : list of strings \n list of transforms generated by ants.registration where each transform is a filename.\n \n interpolator : string \n Choice of interpolator. Supports partial matching.\n linear\n nearestNeighbor\n multiLabel for label images but genericlabel is preferred\n gaussian\n bSpline\n cosineWindowedSinc\n welchWindowedSinc\n hammingWindowedSinc\n lanczosWindowedSinc\n genericLabel use this for label images\n \n imagetype : integer \n choose 0/1/2/3 mapping to scalar/vector/tensor/time-series\n \n whichtoinvert : list of booleans (optional) \n Must be same length as transformlist.\n whichtoinvert[i] is True if transformlist[i] is a matrix, \n and the matrix should be inverted. If transformlist[i] is a \n warp field, whichtoinvert[i] must be False.\n If the transform list is a matrix followed by a warp field, \n whichtoinvert defaults to (True,False). Otherwise it defaults \n to [False]*len(transformlist)).\n \n compose : string (optional)\n if it is a string pointing to a valid file location, \n this will force the function to return a composite transformation filename.\n \n verbose : boolean\n print command and run verbose application of transform.\n \n kwargs : keyword arguments \n extra parameters\n \n Returns\n -------\n ANTsImage or string (transformation filename)\n\n Example\n -------\n >>> import ants\n >>> fixed = ants.image_read( ants.get_ants_data('r16') )\n >>> moving = ants.image_read( ants.get_ants_data('r64') )\n >>> fixed = ants.resample_image(fixed, (64,64), 1, 0)\n >>> moving = ants.resample_image(moving, (64,64), 1, 0)\n >>> mytx = ants.registration(fixed=fixed , moving=moving ,\n type_of_transform = 'SyN' )\n >>> mywarpedimage = ants.apply_transforms( fixed=fixed, moving=moving,\n transformlist=mytx['fwdtransforms'] )\n "
if ((not isinstance(transformlist, (tuple, list))) and (transformlist is not None)):
transformlist = [transformlist]
accepted_interpolators = {'linear', 'nearestNeighbor', 'multiLabel', 'gaussian', 'bSpline', 'cosineWindowedSinc', 'welchWindowedSinc', 'hammingWindowedSinc', 'lanczosWindowedSinc', 'genericLabel'}
if (interpolator not in accepted_interpolators):
raise ValueError(('interpolator not supported - see %s' % accepted_interpolators))
args = [fixed, moving, transformlist, interpolator]
if (not isinstance(fixed, str)):
if (isinstance(fixed, iio.ANTsImage) and isinstance(moving, iio.ANTsImage)):
for tl_path in transformlist:
if (not os.path.exists(tl_path)):
raise Exception(('Transform %s does not exist' % tl_path))
inpixeltype = fixed.pixeltype
fixed = fixed.clone('float')
moving = moving.clone('float')
warpedmovout = moving.clone()
f = fixed
m = moving
if ((moving.dimension == 4) and (fixed.dimension == 3) and (imagetype == 0)):
raise Exception('Set imagetype 3 to transform time series images.')
wmo = warpedmovout
mytx = []
if ((whichtoinvert is None) or (isinstance(whichtoinvert, (tuple, list)) and (sum([(w is not None) for w in whichtoinvert]) == 0))):
if ((len(transformlist) == 2) and ('.mat' in transformlist[0]) and ('.mat' not in transformlist[1])):
whichtoinvert = (True, False)
else:
whichtoinvert = tuple(([False] * len(transformlist)))
if (len(whichtoinvert) != len(transformlist)):
raise ValueError('Transform list and inversion list must be the same length')
for i in range(len(transformlist)):
ismat = False
if ('.mat' in transformlist[0]):
ismat = True
if (whichtoinvert[i] and (not ismat)):
raise ValueError(('Cannot invert transform %i (%s) because it is not a matrix' % (i, transformlist[i])))
if whichtoinvert[i]:
mytx = (mytx + ['-t', ('[%s,1]' % transformlist[i])])
else:
mytx = (mytx + ['-t', transformlist[i]])
if (compose is None):
args = ['-d', fixed.dimension, '-i', m, '-o', wmo, '-r', f, '-n', interpolator]
args = (args + mytx)
tfn = (('%scomptx.nii.gz' % compose) if (compose is not None) else 'NA')
if (compose is not None):
mycompo = ('[%s,1]' % tfn)
args = ['-d', fixed.dimension, '-i', m, '-o', mycompo, '-r', f, '-n', interpolator]
args = (args + mytx)
myargs = utils._int_antsProcessArguments(args)
for jj in range(len(myargs)):
if (myargs[jj] is not None):
if (myargs[jj] == '-'):
myargs2 = ([None] * (len(myargs) - 1))
myargs2[:(jj - 1)] = myargs[:(jj - 1)]
myargs2[jj:(len(myargs) - 1)] = myargs[(jj + 1):len(myargs)]
myargs = myargs2
myverb = int(verbose)
if verbose:
print(myargs)
processed_args = (myargs + ['-z', str(1), '-v', str(myverb), '--float', str(1), '-e', str(imagetype)])
libfn = utils.get_lib_fn('antsApplyTransforms')
libfn(processed_args)
if (compose is None):
return warpedmovout.clone(inpixeltype)
elif os.path.exists(tfn):
return tfn
else:
return None
else:
return 1
else:
args = (args + ['-z', 1, '--float', 1, '-e', imagetype])
processed_args = utils._int_antsProcessArguments(args)
libfn = utils.get_lib_fn('antsApplyTransforms')
libfn(processed_args)<|docstring|>Apply a transform list to map an image from one domain to another.
In image registration, one computes mappings between (usually) pairs
of images. These transforms are often a sequence of increasingly
complex maps, e.g. from translation, to rigid, to affine to deformation.
The list of such transforms is passed to this function to interpolate one
image domain into the next image domain, as below. The order matters
strongly and the user is advised to familiarize with the standards
established in examples.
ANTsR function: `antsApplyTransforms`
Arguments
---------
fixed : ANTsImage
fixed image defining domain into which the moving image is transformed.
moving : AntsImage
moving image to be mapped to fixed space.
transformlist : list of strings
list of transforms generated by ants.registration where each transform is a filename.
interpolator : string
Choice of interpolator. Supports partial matching.
linear
nearestNeighbor
multiLabel for label images but genericlabel is preferred
gaussian
bSpline
cosineWindowedSinc
welchWindowedSinc
hammingWindowedSinc
lanczosWindowedSinc
genericLabel use this for label images
imagetype : integer
choose 0/1/2/3 mapping to scalar/vector/tensor/time-series
whichtoinvert : list of booleans (optional)
Must be same length as transformlist.
whichtoinvert[i] is True if transformlist[i] is a matrix,
and the matrix should be inverted. If transformlist[i] is a
warp field, whichtoinvert[i] must be False.
If the transform list is a matrix followed by a warp field,
whichtoinvert defaults to (True,False). Otherwise it defaults
to [False]*len(transformlist)).
compose : string (optional)
if it is a string pointing to a valid file location,
this will force the function to return a composite transformation filename.
verbose : boolean
print command and run verbose application of transform.
kwargs : keyword arguments
extra parameters
Returns
-------
ANTsImage or string (transformation filename)
Example
-------
>>> import ants
>>> fixed = ants.image_read( ants.get_ants_data('r16') )
>>> moving = ants.image_read( ants.get_ants_data('r64') )
>>> fixed = ants.resample_image(fixed, (64,64), 1, 0)
>>> moving = ants.resample_image(moving, (64,64), 1, 0)
>>> mytx = ants.registration(fixed=fixed , moving=moving ,
type_of_transform = 'SyN' )
>>> mywarpedimage = ants.apply_transforms( fixed=fixed, moving=moving,
transformlist=mytx['fwdtransforms'] )<|endoftext|>
|
3b6eecd96d14de07c16d3a4b46d410afd9728c2fc73a050062045ed58602f261
|
def test20(self):
'SDT()'
self.assertEqual(repr(SDT()), 'SDT()')
|
SDT()
|
sdt_metrics/tests/test__sdt_metrics.py
|
test20
|
rogerlew/sdt-metrics
| 2 |
python
|
def test20(self):
self.assertEqual(repr(), )
|
def test20(self):
self.assertEqual(repr(), )<|docstring|>SDT()<|endoftext|>
|
1904fd3e9bc65fa2aaf3373fa5754a042ba2dcdf6c9146077f410687d7d39bee
|
def test21(self):
'SDT(mapping)'
D = SDT(dict([(HI, 10), (CR, 9), (MI, 1)]))
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')
|
SDT(mapping)
|
sdt_metrics/tests/test__sdt_metrics.py
|
test21
|
rogerlew/sdt-metrics
| 2 |
python
|
def test21(self):
D = SDT(dict([(HI, 10), (CR, 9), (MI, 1)]))
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')
|
def test21(self):
D = SDT(dict([(HI, 10), (CR, 9), (MI, 1)]))
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')<|docstring|>SDT(mapping)<|endoftext|>
|
e078e4f46b80d11de6632d6d740d998bb90b0ceca2aba7521436911e081d6052
|
def test22(self):
'SDT(iterable)'
D = SDT([(HI, 10), (CR, 9), (MI, 1)])
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')
|
SDT(iterable)
|
sdt_metrics/tests/test__sdt_metrics.py
|
test22
|
rogerlew/sdt-metrics
| 2 |
python
|
def test22(self):
D = SDT([(HI, 10), (CR, 9), (MI, 1)])
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')
|
def test22(self):
D = SDT([(HI, 10), (CR, 9), (MI, 1)])
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')<|docstring|>SDT(iterable)<|endoftext|>
|
95dcf4aae0ca5f4e15bf1a2cd109224636fd26beceb83661807fb2f9505286b9
|
def test23(self):
'SDT(**kwargs)'
D = SDT(HI=10, MI=1, CR=9)
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')
|
SDT(**kwargs)
|
sdt_metrics/tests/test__sdt_metrics.py
|
test23
|
rogerlew/sdt-metrics
| 2 |
python
|
def test23(self):
D = SDT(HI=10, MI=1, CR=9)
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')
|
def test23(self):
D = SDT(HI=10, MI=1, CR=9)
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')<|docstring|>SDT(**kwargs)<|endoftext|>
|
4db1f6ea0a44fb00d95fa3363fa85125ef6191183c878ef5f37af77cf69565a1
|
def test24(self):
'SDT(iterable)'
D = SDT([HI, HI, HI, FA, FA])
self.assertEqual(repr(D), 'SDT(HI=3, MI=0, CR=0, FA=2)')
|
SDT(iterable)
|
sdt_metrics/tests/test__sdt_metrics.py
|
test24
|
rogerlew/sdt-metrics
| 2 |
python
|
def test24(self):
D = SDT([HI, HI, HI, FA, FA])
self.assertEqual(repr(D), 'SDT(HI=3, MI=0, CR=0, FA=2)')
|
def test24(self):
D = SDT([HI, HI, HI, FA, FA])
self.assertEqual(repr(D), 'SDT(HI=3, MI=0, CR=0, FA=2)')<|docstring|>SDT(iterable)<|endoftext|>
|
cdc363b61d1a30a3e88b7755def46f816399023077b14eecc506e00b345003db
|
def test25(self):
'SDT(iterable, **kwargs), with overlapping key/values'
D = SDT(dict([(HI, 10), (CR, 9)]), MI=1)
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')
|
SDT(iterable, **kwargs), with overlapping key/values
|
sdt_metrics/tests/test__sdt_metrics.py
|
test25
|
rogerlew/sdt-metrics
| 2 |
python
|
def test25(self):
D = SDT(dict([(HI, 10), (CR, 9)]), MI=1)
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')
|
def test25(self):
D = SDT(dict([(HI, 10), (CR, 9)]), MI=1)
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')<|docstring|>SDT(iterable, **kwargs), with overlapping key/values<|endoftext|>
|
4924d66b17067cbf772a722e9e87d62fc89a5ff107e5cd1bc0f3026dc41d5973
|
def test99(self):
'Make sure that direct calls to update\n do not clear previous contents'
D = SDT(dict([(HI, 10), (CR, 9)]))
D.__init__(MI=1)
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')
|
Make sure that direct calls to update
do not clear previous contents
|
sdt_metrics/tests/test__sdt_metrics.py
|
test99
|
rogerlew/sdt-metrics
| 2 |
python
|
def test99(self):
'Make sure that direct calls to update\n do not clear previous contents'
D = SDT(dict([(HI, 10), (CR, 9)]))
D.__init__(MI=1)
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')
|
def test99(self):
'Make sure that direct calls to update\n do not clear previous contents'
D = SDT(dict([(HI, 10), (CR, 9)]))
D.__init__(MI=1)
self.assertEqual(repr(D), 'SDT(HI=10, MI=1, CR=9, FA=0)')<|docstring|>Make sure that direct calls to update
do not clear previous contents<|endoftext|>
|
5e4363c7ca4959fec3766843d20fdc98606761a129240fed0865ea772af9b378
|
def test20(self):
'SDT()'
D = SDT()
D.subtract()
self.assertEqual(repr(D), 'SDT()')
|
SDT()
|
sdt_metrics/tests/test__sdt_metrics.py
|
test20
|
rogerlew/sdt-metrics
| 2 |
python
|
def test20(self):
D =
D.subtract()
self.assertEqual(repr(D), )
|
def test20(self):
D =
D.subtract()
self.assertEqual(repr(D), )<|docstring|>SDT()<|endoftext|>
|
e958e103dbc4e7b72f9e48ed0c3f6e83909d3f00e6ff3c8672a846f918d11153
|
def test21(self):
'SDT(mapping)'
D = SDT()
D.subtract(dict([(HI, 10), (CR, 9), (MI, 1)]))
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')
|
SDT(mapping)
|
sdt_metrics/tests/test__sdt_metrics.py
|
test21
|
rogerlew/sdt-metrics
| 2 |
python
|
def test21(self):
D = SDT()
D.subtract(dict([(HI, 10), (CR, 9), (MI, 1)]))
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')
|
def test21(self):
D = SDT()
D.subtract(dict([(HI, 10), (CR, 9), (MI, 1)]))
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')<|docstring|>SDT(mapping)<|endoftext|>
|
bab3533369b7640ac17b09f51a50688678e7883c891d577ae5b9a38b7cceb87d
|
def test22(self):
'SDT(iterable)'
D = SDT()
D.subtract([(HI, 10), (CR, 9), (MI, 1)])
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')
|
SDT(iterable)
|
sdt_metrics/tests/test__sdt_metrics.py
|
test22
|
rogerlew/sdt-metrics
| 2 |
python
|
def test22(self):
D = SDT()
D.subtract([(HI, 10), (CR, 9), (MI, 1)])
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')
|
def test22(self):
D = SDT()
D.subtract([(HI, 10), (CR, 9), (MI, 1)])
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')<|docstring|>SDT(iterable)<|endoftext|>
|
600605804eb85c19a3fdb9cf966d50491cec4a8e3e462e1b00e57eade840f866
|
def test23(self):
'SDT(**kwargs)'
D = SDT()
D.subtract(HI=10, MI=1, CR=9)
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')
|
SDT(**kwargs)
|
sdt_metrics/tests/test__sdt_metrics.py
|
test23
|
rogerlew/sdt-metrics
| 2 |
python
|
def test23(self):
D = SDT()
D.subtract(HI=10, MI=1, CR=9)
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')
|
def test23(self):
D = SDT()
D.subtract(HI=10, MI=1, CR=9)
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')<|docstring|>SDT(**kwargs)<|endoftext|>
|
ea48e62a2fa84f16d2ca31b6cbc5ef784fe300d2226a05736d2b2ecb2785caca
|
def test25(self):
'SDT(iterable, **kwargs), with overlapping key/values'
D = SDT()
D.subtract(dict([(HI, 10), (CR, 9)]), MI=1)
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')
|
SDT(iterable, **kwargs), with overlapping key/values
|
sdt_metrics/tests/test__sdt_metrics.py
|
test25
|
rogerlew/sdt-metrics
| 2 |
python
|
def test25(self):
D = SDT()
D.subtract(dict([(HI, 10), (CR, 9)]), MI=1)
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')
|
def test25(self):
D = SDT()
D.subtract(dict([(HI, 10), (CR, 9)]), MI=1)
self.assertEqual(repr(D), 'SDT(HI=-10, MI=-1, CR=-9, FA=0)')<|docstring|>SDT(iterable, **kwargs), with overlapping key/values<|endoftext|>
|
393ef8d267cde71d5f97aad8633316130e18743159eb7d71d9d062a2ac2cb378
|
def test26(self):
'SDT(iterable, **kwargs), with overlapping key/values'
D = SDT(dict([(HI, 10), (CR, 9)]), MI=1)
D.update([HI, HI, HI, FA, FA])
self.assertEqual(repr(D), 'SDT(HI=13, MI=1, CR=9, FA=2)')
|
SDT(iterable, **kwargs), with overlapping key/values
|
sdt_metrics/tests/test__sdt_metrics.py
|
test26
|
rogerlew/sdt-metrics
| 2 |
python
|
def test26(self):
D = SDT(dict([(HI, 10), (CR, 9)]), MI=1)
D.update([HI, HI, HI, FA, FA])
self.assertEqual(repr(D), 'SDT(HI=13, MI=1, CR=9, FA=2)')
|
def test26(self):
D = SDT(dict([(HI, 10), (CR, 9)]), MI=1)
D.update([HI, HI, HI, FA, FA])
self.assertEqual(repr(D), 'SDT(HI=13, MI=1, CR=9, FA=2)')<|docstring|>SDT(iterable, **kwargs), with overlapping key/values<|endoftext|>
|
7de3fcadc3fff637536e37770d7da9f67a899cbfdd2b5e1ee32937d8ed5e1b9d
|
def test0(self):
'float args'
self.assertEqual(aprime.direct(12, 3, 4, 34), SDT(HI=12, MI=3, CR=4, FA=34).aprime())
|
float args
|
sdt_metrics/tests/test__sdt_metrics.py
|
test0
|
rogerlew/sdt-metrics
| 2 |
python
|
def test0(self):
self.assertEqual(aprime.direct(12, 3, 4, 34), SDT(HI=12, MI=3, CR=4, FA=34).aprime())
|
def test0(self):
self.assertEqual(aprime.direct(12, 3, 4, 34), SDT(HI=12, MI=3, CR=4, FA=34).aprime())<|docstring|>float args<|endoftext|>
|
f64a71ca38e070cf62296f6d145d16159996a7fc12c39cf87284e77b74f95f22
|
def test1(self):
'list args'
R = [SDT(HI=12, MI=3, CR=4, FA=34).aprime(), SDT(HI=12, MI=3, CR=4, FA=4).aprime()]
D = aprime.direct([12, 12], [3, 3], [4, 4], [34, 4])
for (r, d) in zip(R, D):
self.assertAlmostEqual(r, d, 7)
|
list args
|
sdt_metrics/tests/test__sdt_metrics.py
|
test1
|
rogerlew/sdt-metrics
| 2 |
python
|
def test1(self):
R = [SDT(HI=12, MI=3, CR=4, FA=34).aprime(), SDT(HI=12, MI=3, CR=4, FA=4).aprime()]
D = aprime.direct([12, 12], [3, 3], [4, 4], [34, 4])
for (r, d) in zip(R, D):
self.assertAlmostEqual(r, d, 7)
|
def test1(self):
R = [SDT(HI=12, MI=3, CR=4, FA=34).aprime(), SDT(HI=12, MI=3, CR=4, FA=4).aprime()]
D = aprime.direct([12, 12], [3, 3], [4, 4], [34, 4])
for (r, d) in zip(R, D):
self.assertAlmostEqual(r, d, 7)<|docstring|>list args<|endoftext|>
|
b76ece9fe947b5e9a4bcceee8182f1274ad3f175187848717c10dedb3c25deed
|
def test0(self):
'float args'
self.assertEqual(aprime.prob((12 / 15.0), (34 / 38.0)), SDT(HI=12, MI=3, CR=4, FA=34).aprime())
|
float args
|
sdt_metrics/tests/test__sdt_metrics.py
|
test0
|
rogerlew/sdt-metrics
| 2 |
python
|
def test0(self):
self.assertEqual(aprime.prob((12 / 15.0), (34 / 38.0)), SDT(HI=12, MI=3, CR=4, FA=34).aprime())
|
def test0(self):
self.assertEqual(aprime.prob((12 / 15.0), (34 / 38.0)), SDT(HI=12, MI=3, CR=4, FA=34).aprime())<|docstring|>float args<|endoftext|>
|
445367e7c7e5a7cc5108c0e950709f12575e875d1fe61e7d21544a564fddcb70
|
def test1(self):
'list args'
R = [SDT(HI=12, MI=3, CR=4, FA=34).aprime(), SDT(HI=12, MI=3, CR=4, FA=4).aprime()]
D = aprime.prob([(12 / 15.0), (12 / 15.0)], [(34 / 38.0), (4 / 8.0)])
for (r, d) in zip(R, D):
self.assertAlmostEqual(r, d, 7)
|
list args
|
sdt_metrics/tests/test__sdt_metrics.py
|
test1
|
rogerlew/sdt-metrics
| 2 |
python
|
def test1(self):
R = [SDT(HI=12, MI=3, CR=4, FA=34).aprime(), SDT(HI=12, MI=3, CR=4, FA=4).aprime()]
D = aprime.prob([(12 / 15.0), (12 / 15.0)], [(34 / 38.0), (4 / 8.0)])
for (r, d) in zip(R, D):
self.assertAlmostEqual(r, d, 7)
|
def test1(self):
R = [SDT(HI=12, MI=3, CR=4, FA=34).aprime(), SDT(HI=12, MI=3, CR=4, FA=4).aprime()]
D = aprime.prob([(12 / 15.0), (12 / 15.0)], [(34 / 38.0), (4 / 8.0)])
for (r, d) in zip(R, D):
self.assertAlmostEqual(r, d, 7)<|docstring|>list args<|endoftext|>
|
0f01f87baff1c2295b02819a3fb338fd37e68dc4b5c0a744d55c07a736671e81
|
def test2(self):
'test _prob binding'
self.assertEqual(hasattr(mutual_info, 'prob'), False)
|
test _prob binding
|
sdt_metrics/tests/test__sdt_metrics.py
|
test2
|
rogerlew/sdt-metrics
| 2 |
python
|
def test2(self):
self.assertEqual(hasattr(mutual_info, 'prob'), False)
|
def test2(self):
self.assertEqual(hasattr(mutual_info, 'prob'), False)<|docstring|>test _prob binding<|endoftext|>
|
97bb8e5cab6defa5167afd6c15c566e83c83270b1f125b38726f69295dce90bd
|
def test1(self):
'given an SDT object'
sdt = SDT(HI=116, MI=30, CR=323, FA=80)
sdt_metrics.plotting.poc_plot(sdt)
|
given an SDT object
|
sdt_metrics/tests/test__sdt_metrics.py
|
test1
|
rogerlew/sdt-metrics
| 2 |
python
|
def test1(self):
sdt = SDT(HI=116, MI=30, CR=323, FA=80)
sdt_metrics.plotting.poc_plot(sdt)
|
def test1(self):
sdt = SDT(HI=116, MI=30, CR=323, FA=80)
sdt_metrics.plotting.poc_plot(sdt)<|docstring|>given an SDT object<|endoftext|>
|
c907d11b4da8e48098e1e256c91c9249d16be7b35beea27964c292db0da05952
|
def test2(self):
'given probabilities'
sdt_metrics.plotting.poc_plot(0.67, 0.43)
|
given probabilities
|
sdt_metrics/tests/test__sdt_metrics.py
|
test2
|
rogerlew/sdt-metrics
| 2 |
python
|
def test2(self):
sdt_metrics.plotting.poc_plot(0.67, 0.43)
|
def test2(self):
sdt_metrics.plotting.poc_plot(0.67, 0.43)<|docstring|>given probabilities<|endoftext|>
|
ab626af3acac97cbfa4bf69309753f48937d2fd27e6ada39be25a7c1d6643158
|
def test3(self):
'given an counts'
sdt_metrics.plotting.poc_plot(116, 30, 50, 50)
|
given an counts
|
sdt_metrics/tests/test__sdt_metrics.py
|
test3
|
rogerlew/sdt-metrics
| 2 |
python
|
def test3(self):
sdt_metrics.plotting.poc_plot(116, 30, 50, 50)
|
def test3(self):
sdt_metrics.plotting.poc_plot(116, 30, 50, 50)<|docstring|>given an counts<|endoftext|>
|
b13cc62627a2883989c135beaa582f53085804f055927f574e86346c34d334a9
|
def test1(self):
'given an SDT object'
sdt = SDT(HI=116, MI=30, CR=323, FA=80)
sdt_metrics.plotting.roc_plot(sdt)
|
given an SDT object
|
sdt_metrics/tests/test__sdt_metrics.py
|
test1
|
rogerlew/sdt-metrics
| 2 |
python
|
def test1(self):
sdt = SDT(HI=116, MI=30, CR=323, FA=80)
sdt_metrics.plotting.roc_plot(sdt)
|
def test1(self):
sdt = SDT(HI=116, MI=30, CR=323, FA=80)
sdt_metrics.plotting.roc_plot(sdt)<|docstring|>given an SDT object<|endoftext|>
|
742f835f90b3c4157e6e9aaebaaad3b7ed262d637f6e8fd066edd3a50eb06f98
|
def test2(self):
'given probabilities'
sdt_metrics.plotting.roc_plot(0.67, 0.43, metric='amzs', fname='roc_example01.png')
|
given probabilities
|
sdt_metrics/tests/test__sdt_metrics.py
|
test2
|
rogerlew/sdt-metrics
| 2 |
python
|
def test2(self):
sdt_metrics.plotting.roc_plot(0.67, 0.43, metric='amzs', fname='roc_example01.png')
|
def test2(self):
sdt_metrics.plotting.roc_plot(0.67, 0.43, metric='amzs', fname='roc_example01.png')<|docstring|>given probabilities<|endoftext|>
|
05057432b2c80244f7a197f9900dbef5fe32e22258d7bfd18e8c20aca3af347a
|
def test3(self):
'given an counts'
sdt_metrics.plotting.roc_plot(116, 30, 50, 50, metric='aprime', fname='roc_example02.png')
|
given an counts
|
sdt_metrics/tests/test__sdt_metrics.py
|
test3
|
rogerlew/sdt-metrics
| 2 |
python
|
def test3(self):
sdt_metrics.plotting.roc_plot(116, 30, 50, 50, metric='aprime', fname='roc_example02.png')
|
def test3(self):
sdt_metrics.plotting.roc_plot(116, 30, 50, 50, metric='aprime', fname='roc_example02.png')<|docstring|>given an counts<|endoftext|>
|
af30292a78303bd8cbaaf37785422e0983e4a047fb499a5b6bca0958a38d218b
|
def test4(self):
'given SDT object'
sdt_metrics.plotting.roc_plot(SDT(HI=251, MI=245, CR=264, FA=240), fname='roc_example03.png')
|
given SDT object
|
sdt_metrics/tests/test__sdt_metrics.py
|
test4
|
rogerlew/sdt-metrics
| 2 |
python
|
def test4(self):
sdt_metrics.plotting.roc_plot(SDT(HI=251, MI=245, CR=264, FA=240), fname='roc_example03.png')
|
def test4(self):
sdt_metrics.plotting.roc_plot(SDT(HI=251, MI=245, CR=264, FA=240), fname='roc_example03.png')<|docstring|>given SDT object<|endoftext|>
|
b9fd9ae4cb8094b2da1f5a2fd94d305cff73a3a058b9e58badbc535930853dd2
|
def test4(self):
'given isopleth'
sdt_metrics.plotting.roc_plot(116, 30, 50, 50, metric='dprime', isopleths='beta', fname='roc_example04.png')
|
given isopleth
|
sdt_metrics/tests/test__sdt_metrics.py
|
test4
|
rogerlew/sdt-metrics
| 2 |
python
|
def test4(self):
sdt_metrics.plotting.roc_plot(116, 30, 50, 50, metric='dprime', isopleths='beta', fname='roc_example04.png')
|
def test4(self):
sdt_metrics.plotting.roc_plot(116, 30, 50, 50, metric='dprime', isopleths='beta', fname='roc_example04.png')<|docstring|>given isopleth<|endoftext|>
|
dfbc0678dae84b9615a9f1ec502dd4266f1cac74e57b934e3ec6c53da38241ae
|
def __init__(self, id=None, name=None, type=None, organization=None, created_by=None, created_at=None, local_token=None, automated_project_settings=None):
'ConnectorDto - a model defined in Swagger'
self._id = None
self._name = None
self._type = None
self._organization = None
self._created_by = None
self._created_at = None
self._local_token = None
self._automated_project_settings = None
self.discriminator = None
if (id is not None):
self.id = id
if (name is not None):
self.name = name
if (type is not None):
self.type = type
if (organization is not None):
self.organization = organization
if (created_by is not None):
self.created_by = created_by
if (created_at is not None):
self.created_at = created_at
if (local_token is not None):
self.local_token = local_token
if (automated_project_settings is not None):
self.automated_project_settings = automated_project_settings
|
ConnectorDto - a model defined in Swagger
|
memsource_cli/models/connector_dto.py
|
__init__
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
def __init__(self, id=None, name=None, type=None, organization=None, created_by=None, created_at=None, local_token=None, automated_project_settings=None):
self._id = None
self._name = None
self._type = None
self._organization = None
self._created_by = None
self._created_at = None
self._local_token = None
self._automated_project_settings = None
self.discriminator = None
if (id is not None):
self.id = id
if (name is not None):
self.name = name
if (type is not None):
self.type = type
if (organization is not None):
self.organization = organization
if (created_by is not None):
self.created_by = created_by
if (created_at is not None):
self.created_at = created_at
if (local_token is not None):
self.local_token = local_token
if (automated_project_settings is not None):
self.automated_project_settings = automated_project_settings
|
def __init__(self, id=None, name=None, type=None, organization=None, created_by=None, created_at=None, local_token=None, automated_project_settings=None):
self._id = None
self._name = None
self._type = None
self._organization = None
self._created_by = None
self._created_at = None
self._local_token = None
self._automated_project_settings = None
self.discriminator = None
if (id is not None):
self.id = id
if (name is not None):
self.name = name
if (type is not None):
self.type = type
if (organization is not None):
self.organization = organization
if (created_by is not None):
self.created_by = created_by
if (created_at is not None):
self.created_at = created_at
if (local_token is not None):
self.local_token = local_token
if (automated_project_settings is not None):
self.automated_project_settings = automated_project_settings<|docstring|>ConnectorDto - a model defined in Swagger<|endoftext|>
|
85d936216b6a7f6d4fc446f35eaf5a71c88b5c2991827c7aecb9ebc46e0091d5
|
@property
def id(self):
'Gets the id of this ConnectorDto. # noqa: E501\n\n\n :return: The id of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._id
|
Gets the id of this ConnectorDto. # noqa: E501
:return: The id of this ConnectorDto. # noqa: E501
:rtype: str
|
memsource_cli/models/connector_dto.py
|
id
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@property
def id(self):
'Gets the id of this ConnectorDto. # noqa: E501\n\n\n :return: The id of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._id
|
@property
def id(self):
'Gets the id of this ConnectorDto. # noqa: E501\n\n\n :return: The id of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._id<|docstring|>Gets the id of this ConnectorDto. # noqa: E501
:return: The id of this ConnectorDto. # noqa: E501
:rtype: str<|endoftext|>
|
37f682a3fe7e2b615c1da9c5e64f10ad100abe2f52a26dfffb4b62b42a3e7aa9
|
@id.setter
def id(self, id):
'Sets the id of this ConnectorDto.\n\n\n :param id: The id of this ConnectorDto. # noqa: E501\n :type: str\n '
self._id = id
|
Sets the id of this ConnectorDto.
:param id: The id of this ConnectorDto. # noqa: E501
:type: str
|
memsource_cli/models/connector_dto.py
|
id
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@id.setter
def id(self, id):
'Sets the id of this ConnectorDto.\n\n\n :param id: The id of this ConnectorDto. # noqa: E501\n :type: str\n '
self._id = id
|
@id.setter
def id(self, id):
'Sets the id of this ConnectorDto.\n\n\n :param id: The id of this ConnectorDto. # noqa: E501\n :type: str\n '
self._id = id<|docstring|>Sets the id of this ConnectorDto.
:param id: The id of this ConnectorDto. # noqa: E501
:type: str<|endoftext|>
|
aa1aa50ea5592aa8fc76b3547be9ee2a88108529423db8ba196606ecfc938a8e
|
@property
def name(self):
'Gets the name of this ConnectorDto. # noqa: E501\n\n\n :return: The name of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._name
|
Gets the name of this ConnectorDto. # noqa: E501
:return: The name of this ConnectorDto. # noqa: E501
:rtype: str
|
memsource_cli/models/connector_dto.py
|
name
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@property
def name(self):
'Gets the name of this ConnectorDto. # noqa: E501\n\n\n :return: The name of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._name
|
@property
def name(self):
'Gets the name of this ConnectorDto. # noqa: E501\n\n\n :return: The name of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._name<|docstring|>Gets the name of this ConnectorDto. # noqa: E501
:return: The name of this ConnectorDto. # noqa: E501
:rtype: str<|endoftext|>
|
e3472001f18ab69697a558b5183bf19067b7b03a80017c1f6af6f4c9e47b675e
|
@name.setter
def name(self, name):
'Sets the name of this ConnectorDto.\n\n\n :param name: The name of this ConnectorDto. # noqa: E501\n :type: str\n '
self._name = name
|
Sets the name of this ConnectorDto.
:param name: The name of this ConnectorDto. # noqa: E501
:type: str
|
memsource_cli/models/connector_dto.py
|
name
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@name.setter
def name(self, name):
'Sets the name of this ConnectorDto.\n\n\n :param name: The name of this ConnectorDto. # noqa: E501\n :type: str\n '
self._name = name
|
@name.setter
def name(self, name):
'Sets the name of this ConnectorDto.\n\n\n :param name: The name of this ConnectorDto. # noqa: E501\n :type: str\n '
self._name = name<|docstring|>Sets the name of this ConnectorDto.
:param name: The name of this ConnectorDto. # noqa: E501
:type: str<|endoftext|>
|
2dd48d2efbf834caef36584364cf91faf07c2de095e9c69815f20ddc6f82b0d9
|
@property
def type(self):
'Gets the type of this ConnectorDto. # noqa: E501\n\n\n :return: The type of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._type
|
Gets the type of this ConnectorDto. # noqa: E501
:return: The type of this ConnectorDto. # noqa: E501
:rtype: str
|
memsource_cli/models/connector_dto.py
|
type
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@property
def type(self):
'Gets the type of this ConnectorDto. # noqa: E501\n\n\n :return: The type of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._type
|
@property
def type(self):
'Gets the type of this ConnectorDto. # noqa: E501\n\n\n :return: The type of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._type<|docstring|>Gets the type of this ConnectorDto. # noqa: E501
:return: The type of this ConnectorDto. # noqa: E501
:rtype: str<|endoftext|>
|
44afd1a53184faca01cac11d63894ba37bc0785f9940117e0ccf02114873f715
|
@type.setter
def type(self, type):
'Sets the type of this ConnectorDto.\n\n\n :param type: The type of this ConnectorDto. # noqa: E501\n :type: str\n '
allowed_values = ['DROPBOX', 'GOOGLE', 'FTP', 'WORDPRESS', 'GITHUB', 'SFTP', 'DRUPAL', 'BOX', 'GIT', 'ZENDESK', 'ONEDRIVE', 'GITLAB', 'MARKETO', 'HUBSPOT', 'HELPSCOUT', 'SALESFORCE', 'BITBUCKET', 'BITBUCKETSERVER', 'SHAREPOINT', 'AZURE', 'SITECORE', 'KENTICO', 'MAGENTO', 'CONTENTFULENTRYLEVEL', 'CONTENTFUL', 'CONTENTSTACK', 'JOOMLA', 'CONFLUENCE', 'TYPO3', 'AEM_PLUGIN']
if (type not in allowed_values):
raise ValueError('Invalid value for `type` ({0}), must be one of {1}'.format(type, allowed_values))
self._type = type
|
Sets the type of this ConnectorDto.
:param type: The type of this ConnectorDto. # noqa: E501
:type: str
|
memsource_cli/models/connector_dto.py
|
type
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@type.setter
def type(self, type):
'Sets the type of this ConnectorDto.\n\n\n :param type: The type of this ConnectorDto. # noqa: E501\n :type: str\n '
allowed_values = ['DROPBOX', 'GOOGLE', 'FTP', 'WORDPRESS', 'GITHUB', 'SFTP', 'DRUPAL', 'BOX', 'GIT', 'ZENDESK', 'ONEDRIVE', 'GITLAB', 'MARKETO', 'HUBSPOT', 'HELPSCOUT', 'SALESFORCE', 'BITBUCKET', 'BITBUCKETSERVER', 'SHAREPOINT', 'AZURE', 'SITECORE', 'KENTICO', 'MAGENTO', 'CONTENTFULENTRYLEVEL', 'CONTENTFUL', 'CONTENTSTACK', 'JOOMLA', 'CONFLUENCE', 'TYPO3', 'AEM_PLUGIN']
if (type not in allowed_values):
raise ValueError('Invalid value for `type` ({0}), must be one of {1}'.format(type, allowed_values))
self._type = type
|
@type.setter
def type(self, type):
'Sets the type of this ConnectorDto.\n\n\n :param type: The type of this ConnectorDto. # noqa: E501\n :type: str\n '
allowed_values = ['DROPBOX', 'GOOGLE', 'FTP', 'WORDPRESS', 'GITHUB', 'SFTP', 'DRUPAL', 'BOX', 'GIT', 'ZENDESK', 'ONEDRIVE', 'GITLAB', 'MARKETO', 'HUBSPOT', 'HELPSCOUT', 'SALESFORCE', 'BITBUCKET', 'BITBUCKETSERVER', 'SHAREPOINT', 'AZURE', 'SITECORE', 'KENTICO', 'MAGENTO', 'CONTENTFULENTRYLEVEL', 'CONTENTFUL', 'CONTENTSTACK', 'JOOMLA', 'CONFLUENCE', 'TYPO3', 'AEM_PLUGIN']
if (type not in allowed_values):
raise ValueError('Invalid value for `type` ({0}), must be one of {1}'.format(type, allowed_values))
self._type = type<|docstring|>Sets the type of this ConnectorDto.
:param type: The type of this ConnectorDto. # noqa: E501
:type: str<|endoftext|>
|
d79e7be45dd519625d9ef04ad97ea4e01fc7fc415e0ea68478e67a3f1b3fa13c
|
@property
def organization(self):
'Gets the organization of this ConnectorDto. # noqa: E501\n\n\n :return: The organization of this ConnectorDto. # noqa: E501\n :rtype: NameDto\n '
return self._organization
|
Gets the organization of this ConnectorDto. # noqa: E501
:return: The organization of this ConnectorDto. # noqa: E501
:rtype: NameDto
|
memsource_cli/models/connector_dto.py
|
organization
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@property
def organization(self):
'Gets the organization of this ConnectorDto. # noqa: E501\n\n\n :return: The organization of this ConnectorDto. # noqa: E501\n :rtype: NameDto\n '
return self._organization
|
@property
def organization(self):
'Gets the organization of this ConnectorDto. # noqa: E501\n\n\n :return: The organization of this ConnectorDto. # noqa: E501\n :rtype: NameDto\n '
return self._organization<|docstring|>Gets the organization of this ConnectorDto. # noqa: E501
:return: The organization of this ConnectorDto. # noqa: E501
:rtype: NameDto<|endoftext|>
|
5aff6e70abfff84fbe99bb754e3d536f64a573b17447d7438ace77cecc0fc5c3
|
@organization.setter
def organization(self, organization):
'Sets the organization of this ConnectorDto.\n\n\n :param organization: The organization of this ConnectorDto. # noqa: E501\n :type: NameDto\n '
self._organization = organization
|
Sets the organization of this ConnectorDto.
:param organization: The organization of this ConnectorDto. # noqa: E501
:type: NameDto
|
memsource_cli/models/connector_dto.py
|
organization
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@organization.setter
def organization(self, organization):
'Sets the organization of this ConnectorDto.\n\n\n :param organization: The organization of this ConnectorDto. # noqa: E501\n :type: NameDto\n '
self._organization = organization
|
@organization.setter
def organization(self, organization):
'Sets the organization of this ConnectorDto.\n\n\n :param organization: The organization of this ConnectorDto. # noqa: E501\n :type: NameDto\n '
self._organization = organization<|docstring|>Sets the organization of this ConnectorDto.
:param organization: The organization of this ConnectorDto. # noqa: E501
:type: NameDto<|endoftext|>
|
d47e5881338849b78b9fa7c227b996826e187e106962b0d1874b9632058ed82b
|
@property
def created_by(self):
'Gets the created_by of this ConnectorDto. # noqa: E501\n\n\n :return: The created_by of this ConnectorDto. # noqa: E501\n :rtype: NameDto\n '
return self._created_by
|
Gets the created_by of this ConnectorDto. # noqa: E501
:return: The created_by of this ConnectorDto. # noqa: E501
:rtype: NameDto
|
memsource_cli/models/connector_dto.py
|
created_by
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@property
def created_by(self):
'Gets the created_by of this ConnectorDto. # noqa: E501\n\n\n :return: The created_by of this ConnectorDto. # noqa: E501\n :rtype: NameDto\n '
return self._created_by
|
@property
def created_by(self):
'Gets the created_by of this ConnectorDto. # noqa: E501\n\n\n :return: The created_by of this ConnectorDto. # noqa: E501\n :rtype: NameDto\n '
return self._created_by<|docstring|>Gets the created_by of this ConnectorDto. # noqa: E501
:return: The created_by of this ConnectorDto. # noqa: E501
:rtype: NameDto<|endoftext|>
|
e9376bf01ad5ba4f03aff8ca0db7a6d2609d6b43b755f81c3142a40d61b7dcc6
|
@created_by.setter
def created_by(self, created_by):
'Sets the created_by of this ConnectorDto.\n\n\n :param created_by: The created_by of this ConnectorDto. # noqa: E501\n :type: NameDto\n '
self._created_by = created_by
|
Sets the created_by of this ConnectorDto.
:param created_by: The created_by of this ConnectorDto. # noqa: E501
:type: NameDto
|
memsource_cli/models/connector_dto.py
|
created_by
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@created_by.setter
def created_by(self, created_by):
'Sets the created_by of this ConnectorDto.\n\n\n :param created_by: The created_by of this ConnectorDto. # noqa: E501\n :type: NameDto\n '
self._created_by = created_by
|
@created_by.setter
def created_by(self, created_by):
'Sets the created_by of this ConnectorDto.\n\n\n :param created_by: The created_by of this ConnectorDto. # noqa: E501\n :type: NameDto\n '
self._created_by = created_by<|docstring|>Sets the created_by of this ConnectorDto.
:param created_by: The created_by of this ConnectorDto. # noqa: E501
:type: NameDto<|endoftext|>
|
86fb75df521c3aa1df8b53dc1544c678c8ce1b50bd2ce5cb382e8049ab380a20
|
@property
def created_at(self):
'Gets the created_at of this ConnectorDto. # noqa: E501\n\n\n :return: The created_at of this ConnectorDto. # noqa: E501\n :rtype: datetime\n '
return self._created_at
|
Gets the created_at of this ConnectorDto. # noqa: E501
:return: The created_at of this ConnectorDto. # noqa: E501
:rtype: datetime
|
memsource_cli/models/connector_dto.py
|
created_at
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@property
def created_at(self):
'Gets the created_at of this ConnectorDto. # noqa: E501\n\n\n :return: The created_at of this ConnectorDto. # noqa: E501\n :rtype: datetime\n '
return self._created_at
|
@property
def created_at(self):
'Gets the created_at of this ConnectorDto. # noqa: E501\n\n\n :return: The created_at of this ConnectorDto. # noqa: E501\n :rtype: datetime\n '
return self._created_at<|docstring|>Gets the created_at of this ConnectorDto. # noqa: E501
:return: The created_at of this ConnectorDto. # noqa: E501
:rtype: datetime<|endoftext|>
|
01c7a38ab9e7ec71d7a10331cba134aac753d6a8a0d30bcdd60f4e610720370f
|
@created_at.setter
def created_at(self, created_at):
'Sets the created_at of this ConnectorDto.\n\n\n :param created_at: The created_at of this ConnectorDto. # noqa: E501\n :type: datetime\n '
self._created_at = created_at
|
Sets the created_at of this ConnectorDto.
:param created_at: The created_at of this ConnectorDto. # noqa: E501
:type: datetime
|
memsource_cli/models/connector_dto.py
|
created_at
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@created_at.setter
def created_at(self, created_at):
'Sets the created_at of this ConnectorDto.\n\n\n :param created_at: The created_at of this ConnectorDto. # noqa: E501\n :type: datetime\n '
self._created_at = created_at
|
@created_at.setter
def created_at(self, created_at):
'Sets the created_at of this ConnectorDto.\n\n\n :param created_at: The created_at of this ConnectorDto. # noqa: E501\n :type: datetime\n '
self._created_at = created_at<|docstring|>Sets the created_at of this ConnectorDto.
:param created_at: The created_at of this ConnectorDto. # noqa: E501
:type: datetime<|endoftext|>
|
b2736258839602b30aed4fe44038af8f263c828a2418cef44638a690f53fe84d
|
@property
def local_token(self):
'Gets the local_token of this ConnectorDto. # noqa: E501\n\n\n :return: The local_token of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._local_token
|
Gets the local_token of this ConnectorDto. # noqa: E501
:return: The local_token of this ConnectorDto. # noqa: E501
:rtype: str
|
memsource_cli/models/connector_dto.py
|
local_token
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@property
def local_token(self):
'Gets the local_token of this ConnectorDto. # noqa: E501\n\n\n :return: The local_token of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._local_token
|
@property
def local_token(self):
'Gets the local_token of this ConnectorDto. # noqa: E501\n\n\n :return: The local_token of this ConnectorDto. # noqa: E501\n :rtype: str\n '
return self._local_token<|docstring|>Gets the local_token of this ConnectorDto. # noqa: E501
:return: The local_token of this ConnectorDto. # noqa: E501
:rtype: str<|endoftext|>
|
46e28415729144c8746c218f7dfc9c82628f6a990fec9e8d7df9333780498cde
|
@local_token.setter
def local_token(self, local_token):
'Sets the local_token of this ConnectorDto.\n\n\n :param local_token: The local_token of this ConnectorDto. # noqa: E501\n :type: str\n '
self._local_token = local_token
|
Sets the local_token of this ConnectorDto.
:param local_token: The local_token of this ConnectorDto. # noqa: E501
:type: str
|
memsource_cli/models/connector_dto.py
|
local_token
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@local_token.setter
def local_token(self, local_token):
'Sets the local_token of this ConnectorDto.\n\n\n :param local_token: The local_token of this ConnectorDto. # noqa: E501\n :type: str\n '
self._local_token = local_token
|
@local_token.setter
def local_token(self, local_token):
'Sets the local_token of this ConnectorDto.\n\n\n :param local_token: The local_token of this ConnectorDto. # noqa: E501\n :type: str\n '
self._local_token = local_token<|docstring|>Sets the local_token of this ConnectorDto.
:param local_token: The local_token of this ConnectorDto. # noqa: E501
:type: str<|endoftext|>
|
21895dda28d9bf2b0fbde1fb879a76797495179976edff3b18fec22c4446332e
|
@property
def automated_project_settings(self):
'Gets the automated_project_settings of this ConnectorDto. # noqa: E501\n\n\n :return: The automated_project_settings of this ConnectorDto. # noqa: E501\n :rtype: list[AutomatedProjectSettingsDto]\n '
return self._automated_project_settings
|
Gets the automated_project_settings of this ConnectorDto. # noqa: E501
:return: The automated_project_settings of this ConnectorDto. # noqa: E501
:rtype: list[AutomatedProjectSettingsDto]
|
memsource_cli/models/connector_dto.py
|
automated_project_settings
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@property
def automated_project_settings(self):
'Gets the automated_project_settings of this ConnectorDto. # noqa: E501\n\n\n :return: The automated_project_settings of this ConnectorDto. # noqa: E501\n :rtype: list[AutomatedProjectSettingsDto]\n '
return self._automated_project_settings
|
@property
def automated_project_settings(self):
'Gets the automated_project_settings of this ConnectorDto. # noqa: E501\n\n\n :return: The automated_project_settings of this ConnectorDto. # noqa: E501\n :rtype: list[AutomatedProjectSettingsDto]\n '
return self._automated_project_settings<|docstring|>Gets the automated_project_settings of this ConnectorDto. # noqa: E501
:return: The automated_project_settings of this ConnectorDto. # noqa: E501
:rtype: list[AutomatedProjectSettingsDto]<|endoftext|>
|
9bf1205f6223204b13356f665b825049d5e8eae32659d71c4436538b3ebccc3b
|
@automated_project_settings.setter
def automated_project_settings(self, automated_project_settings):
'Sets the automated_project_settings of this ConnectorDto.\n\n\n :param automated_project_settings: The automated_project_settings of this ConnectorDto. # noqa: E501\n :type: list[AutomatedProjectSettingsDto]\n '
self._automated_project_settings = automated_project_settings
|
Sets the automated_project_settings of this ConnectorDto.
:param automated_project_settings: The automated_project_settings of this ConnectorDto. # noqa: E501
:type: list[AutomatedProjectSettingsDto]
|
memsource_cli/models/connector_dto.py
|
automated_project_settings
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
@automated_project_settings.setter
def automated_project_settings(self, automated_project_settings):
'Sets the automated_project_settings of this ConnectorDto.\n\n\n :param automated_project_settings: The automated_project_settings of this ConnectorDto. # noqa: E501\n :type: list[AutomatedProjectSettingsDto]\n '
self._automated_project_settings = automated_project_settings
|
@automated_project_settings.setter
def automated_project_settings(self, automated_project_settings):
'Sets the automated_project_settings of this ConnectorDto.\n\n\n :param automated_project_settings: The automated_project_settings of this ConnectorDto. # noqa: E501\n :type: list[AutomatedProjectSettingsDto]\n '
self._automated_project_settings = automated_project_settings<|docstring|>Sets the automated_project_settings of this ConnectorDto.
:param automated_project_settings: The automated_project_settings of this ConnectorDto. # noqa: E501
:type: list[AutomatedProjectSettingsDto]<|endoftext|>
|
2c655fe2921c5f86981218a73b2cb99a72afc68c2a2352dc3e9127f441042a1a
|
def to_dict(self):
'Returns the model properties as a dict'
result = {}
for (attr, _) in six.iteritems(self.swagger_types):
value = getattr(self, attr)
if isinstance(value, list):
result[attr] = list(map((lambda x: (x.to_dict() if hasattr(x, 'to_dict') else x)), value))
elif hasattr(value, 'to_dict'):
result[attr] = value.to_dict()
elif isinstance(value, dict):
result[attr] = dict(map((lambda item: ((item[0], item[1].to_dict()) if hasattr(item[1], 'to_dict') else item)), value.items()))
else:
result[attr] = value
if issubclass(ConnectorDto, dict):
for (key, value) in self.items():
result[key] = value
return result
|
Returns the model properties as a dict
|
memsource_cli/models/connector_dto.py
|
to_dict
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
def to_dict(self):
result = {}
for (attr, _) in six.iteritems(self.swagger_types):
value = getattr(self, attr)
if isinstance(value, list):
result[attr] = list(map((lambda x: (x.to_dict() if hasattr(x, 'to_dict') else x)), value))
elif hasattr(value, 'to_dict'):
result[attr] = value.to_dict()
elif isinstance(value, dict):
result[attr] = dict(map((lambda item: ((item[0], item[1].to_dict()) if hasattr(item[1], 'to_dict') else item)), value.items()))
else:
result[attr] = value
if issubclass(ConnectorDto, dict):
for (key, value) in self.items():
result[key] = value
return result
|
def to_dict(self):
result = {}
for (attr, _) in six.iteritems(self.swagger_types):
value = getattr(self, attr)
if isinstance(value, list):
result[attr] = list(map((lambda x: (x.to_dict() if hasattr(x, 'to_dict') else x)), value))
elif hasattr(value, 'to_dict'):
result[attr] = value.to_dict()
elif isinstance(value, dict):
result[attr] = dict(map((lambda item: ((item[0], item[1].to_dict()) if hasattr(item[1], 'to_dict') else item)), value.items()))
else:
result[attr] = value
if issubclass(ConnectorDto, dict):
for (key, value) in self.items():
result[key] = value
return result<|docstring|>Returns the model properties as a dict<|endoftext|>
|
cbb19eaa2fc8a113d9e32f924ef280a7e97563f8915f94f65dab438997af2e99
|
def to_str(self):
'Returns the string representation of the model'
return pprint.pformat(self.to_dict())
|
Returns the string representation of the model
|
memsource_cli/models/connector_dto.py
|
to_str
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
def to_str(self):
return pprint.pformat(self.to_dict())
|
def to_str(self):
return pprint.pformat(self.to_dict())<|docstring|>Returns the string representation of the model<|endoftext|>
|
772243a2c2b3261a9b954d07aaf295e3c1242a579a495e2d6a5679c677861703
|
def __repr__(self):
'For `print` and `pprint`'
return self.to_str()
|
For `print` and `pprint`
|
memsource_cli/models/connector_dto.py
|
__repr__
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
def __repr__(self):
return self.to_str()
|
def __repr__(self):
return self.to_str()<|docstring|>For `print` and `pprint`<|endoftext|>
|
435fdeec4df2b5adea52e4d3ed64c08491b48301df82ffda98f7a359c5cc739f
|
def __eq__(self, other):
'Returns true if both objects are equal'
if (not isinstance(other, ConnectorDto)):
return False
return (self.__dict__ == other.__dict__)
|
Returns true if both objects are equal
|
memsource_cli/models/connector_dto.py
|
__eq__
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
def __eq__(self, other):
if (not isinstance(other, ConnectorDto)):
return False
return (self.__dict__ == other.__dict__)
|
def __eq__(self, other):
if (not isinstance(other, ConnectorDto)):
return False
return (self.__dict__ == other.__dict__)<|docstring|>Returns true if both objects are equal<|endoftext|>
|
43dc6740163eb9fc1161d09cb2208a64c7ad0cc8d9c8637ac3264522d3ec7e42
|
def __ne__(self, other):
'Returns true if both objects are not equal'
return (not (self == other))
|
Returns true if both objects are not equal
|
memsource_cli/models/connector_dto.py
|
__ne__
|
unofficial-memsource/memsource-cli-client
| 16 |
python
|
def __ne__(self, other):
return (not (self == other))
|
def __ne__(self, other):
return (not (self == other))<|docstring|>Returns true if both objects are not equal<|endoftext|>
|
5875f1c57b381bfd56dd7ae36b3e08ebf9500797cc74a144c88859ad7a03d889
|
def function(arg1, arg2, kwarg1=1):
'_summary_\n\n :param arg1: _description_\n :type arg1: _type_\n :param arg2: _description_\n :type arg2: _type_\n :param kwarg1: _description_, defaults to 1\n :type kwarg1: int, optional\n :raises FileExistsError: _description_\n :return: _description_\n :rtype: _type_\n :yield: _description_\n :rtype: _type_\n '
if (arg2 > 1):
raise FileExistsError()
(yield 1)
return arg1
|
_summary_
:param arg1: _description_
:type arg1: _type_
:param arg2: _description_
:type arg2: _type_
:param kwarg1: _description_, defaults to 1
:type kwarg1: int, optional
:raises FileExistsError: _description_
:return: _description_
:rtype: _type_
:yield: _description_
:rtype: _type_
|
src/test/integration/python_test_files/file_1_output.py
|
function
|
sv1990/autoDocstring
| 0 |
python
|
def function(arg1, arg2, kwarg1=1):
'_summary_\n\n :param arg1: _description_\n :type arg1: _type_\n :param arg2: _description_\n :type arg2: _type_\n :param kwarg1: _description_, defaults to 1\n :type kwarg1: int, optional\n :raises FileExistsError: _description_\n :return: _description_\n :rtype: _type_\n :yield: _description_\n :rtype: _type_\n '
if (arg2 > 1):
raise FileExistsError()
(yield 1)
return arg1
|
def function(arg1, arg2, kwarg1=1):
'_summary_\n\n :param arg1: _description_\n :type arg1: _type_\n :param arg2: _description_\n :type arg2: _type_\n :param kwarg1: _description_, defaults to 1\n :type kwarg1: int, optional\n :raises FileExistsError: _description_\n :return: _description_\n :rtype: _type_\n :yield: _description_\n :rtype: _type_\n '
if (arg2 > 1):
raise FileExistsError()
(yield 1)
return arg1<|docstring|>_summary_
:param arg1: _description_
:type arg1: _type_
:param arg2: _description_
:type arg2: _type_
:param kwarg1: _description_, defaults to 1
:type kwarg1: int, optional
:raises FileExistsError: _description_
:return: _description_
:rtype: _type_
:yield: _description_
:rtype: _type_<|endoftext|>
|
57815e4d7fbf9ffb2b40779300fedb15498d5026a1d3608f964264019fb0101e
|
def SymmetricFunctionAlgebra(R, basis='schur'):
"\n This is deprecated in :trac:`15473`. Use instead\n :class:`SymmetricFunctions` as ``SymmetricFunctions(R).basis()``\n\n INPUT:\n\n - ``R`` -- ring with identity basis\n - ``basis`` -- a string for the name of the basis, must be one of\n 'schur', 'elementary', 'homogeneous', 'power', 'monomial' or their\n abbreviations 's', 'e', 'h', 'p', 'm'\n\n OUTPUT: A SymmetricFunctionAlgebra\n\n EXAMPLES::\n\n sage: SymmetricFunctionAlgebra(QQ)\n doctest:...: DeprecationWarning: this function is deprecated. Use SymmetricFunctions(R).basis()\n See http://trac.sagemath.org/15473 for details.\n Symmetric Functions over Rational Field in the Schur basis\n "
sage.misc.superseded.deprecation(15473, 'this function is deprecated. Use SymmetricFunctions(R).basis()')
from sage.combinat.sf.sf import SymmetricFunctions
Sym = SymmetricFunctions(R)
if ((basis == 'schur') or (basis == 's')):
return Sym.s()
elif ((basis == 'elementary') or (basis == 'e')):
return Sym.e()
elif ((basis == 'homogeneous') or (basis == 'h')):
return Sym.h()
elif ((basis == 'power') or (basis == 'p')):
return Sym.p()
elif ((basis == 'monomial') or (basis == 'm')):
return Sym.m()
else:
raise ValueError(('unknown basis (= %s)' % basis))
|
This is deprecated in :trac:`15473`. Use instead
:class:`SymmetricFunctions` as ``SymmetricFunctions(R).basis()``
INPUT:
- ``R`` -- ring with identity basis
- ``basis`` -- a string for the name of the basis, must be one of
'schur', 'elementary', 'homogeneous', 'power', 'monomial' or their
abbreviations 's', 'e', 'h', 'p', 'm'
OUTPUT: A SymmetricFunctionAlgebra
EXAMPLES::
sage: SymmetricFunctionAlgebra(QQ)
doctest:...: DeprecationWarning: this function is deprecated. Use SymmetricFunctions(R).basis()
See http://trac.sagemath.org/15473 for details.
Symmetric Functions over Rational Field in the Schur basis
|
src/sage/combinat/sf/sfa.py
|
SymmetricFunctionAlgebra
|
bopopescu/sagesmc
| 5 |
python
|
def SymmetricFunctionAlgebra(R, basis='schur'):
"\n This is deprecated in :trac:`15473`. Use instead\n :class:`SymmetricFunctions` as ``SymmetricFunctions(R).basis()``\n\n INPUT:\n\n - ``R`` -- ring with identity basis\n - ``basis`` -- a string for the name of the basis, must be one of\n 'schur', 'elementary', 'homogeneous', 'power', 'monomial' or their\n abbreviations 's', 'e', 'h', 'p', 'm'\n\n OUTPUT: A SymmetricFunctionAlgebra\n\n EXAMPLES::\n\n sage: SymmetricFunctionAlgebra(QQ)\n doctest:...: DeprecationWarning: this function is deprecated. Use SymmetricFunctions(R).basis()\n See http://trac.sagemath.org/15473 for details.\n Symmetric Functions over Rational Field in the Schur basis\n "
sage.misc.superseded.deprecation(15473, 'this function is deprecated. Use SymmetricFunctions(R).basis()')
from sage.combinat.sf.sf import SymmetricFunctions
Sym = SymmetricFunctions(R)
if ((basis == 'schur') or (basis == 's')):
return Sym.s()
elif ((basis == 'elementary') or (basis == 'e')):
return Sym.e()
elif ((basis == 'homogeneous') or (basis == 'h')):
return Sym.h()
elif ((basis == 'power') or (basis == 'p')):
return Sym.p()
elif ((basis == 'monomial') or (basis == 'm')):
return Sym.m()
else:
raise ValueError(('unknown basis (= %s)' % basis))
|
def SymmetricFunctionAlgebra(R, basis='schur'):
"\n This is deprecated in :trac:`15473`. Use instead\n :class:`SymmetricFunctions` as ``SymmetricFunctions(R).basis()``\n\n INPUT:\n\n - ``R`` -- ring with identity basis\n - ``basis`` -- a string for the name of the basis, must be one of\n 'schur', 'elementary', 'homogeneous', 'power', 'monomial' or their\n abbreviations 's', 'e', 'h', 'p', 'm'\n\n OUTPUT: A SymmetricFunctionAlgebra\n\n EXAMPLES::\n\n sage: SymmetricFunctionAlgebra(QQ)\n doctest:...: DeprecationWarning: this function is deprecated. Use SymmetricFunctions(R).basis()\n See http://trac.sagemath.org/15473 for details.\n Symmetric Functions over Rational Field in the Schur basis\n "
sage.misc.superseded.deprecation(15473, 'this function is deprecated. Use SymmetricFunctions(R).basis()')
from sage.combinat.sf.sf import SymmetricFunctions
Sym = SymmetricFunctions(R)
if ((basis == 'schur') or (basis == 's')):
return Sym.s()
elif ((basis == 'elementary') or (basis == 'e')):
return Sym.e()
elif ((basis == 'homogeneous') or (basis == 'h')):
return Sym.h()
elif ((basis == 'power') or (basis == 'p')):
return Sym.p()
elif ((basis == 'monomial') or (basis == 'm')):
return Sym.m()
else:
raise ValueError(('unknown basis (= %s)' % basis))<|docstring|>This is deprecated in :trac:`15473`. Use instead
:class:`SymmetricFunctions` as ``SymmetricFunctions(R).basis()``
INPUT:
- ``R`` -- ring with identity basis
- ``basis`` -- a string for the name of the basis, must be one of
'schur', 'elementary', 'homogeneous', 'power', 'monomial' or their
abbreviations 's', 'e', 'h', 'p', 'm'
OUTPUT: A SymmetricFunctionAlgebra
EXAMPLES::
sage: SymmetricFunctionAlgebra(QQ)
doctest:...: DeprecationWarning: this function is deprecated. Use SymmetricFunctions(R).basis()
See http://trac.sagemath.org/15473 for details.
Symmetric Functions over Rational Field in the Schur basis<|endoftext|>
|
556713846b0ef813fcd72d7f1ee85d3a699ea9934cc113c4745698147f429cab
|
def is_SymmetricFunctionAlgebra(x):
"\n Checks whether ``x`` is a symmetric function algebra.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import is_SymmetricFunctionAlgebra\n sage: is_SymmetricFunctionAlgebra(5)\n False\n sage: is_SymmetricFunctionAlgebra(ZZ)\n False\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(ZZ).schur())\n True\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).e())\n True\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).macdonald(q=1,t=1).P())\n True\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(FractionField(QQ['q','t'])).macdonald().P())\n True\n "
return isinstance(x, SymmetricFunctionAlgebra_generic)
|
Checks whether ``x`` is a symmetric function algebra.
EXAMPLES::
sage: from sage.combinat.sf.sfa import is_SymmetricFunctionAlgebra
sage: is_SymmetricFunctionAlgebra(5)
False
sage: is_SymmetricFunctionAlgebra(ZZ)
False
sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(ZZ).schur())
True
sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).e())
True
sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).macdonald(q=1,t=1).P())
True
sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(FractionField(QQ['q','t'])).macdonald().P())
True
|
src/sage/combinat/sf/sfa.py
|
is_SymmetricFunctionAlgebra
|
bopopescu/sagesmc
| 5 |
python
|
def is_SymmetricFunctionAlgebra(x):
"\n Checks whether ``x`` is a symmetric function algebra.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import is_SymmetricFunctionAlgebra\n sage: is_SymmetricFunctionAlgebra(5)\n False\n sage: is_SymmetricFunctionAlgebra(ZZ)\n False\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(ZZ).schur())\n True\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).e())\n True\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).macdonald(q=1,t=1).P())\n True\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(FractionField(QQ['q','t'])).macdonald().P())\n True\n "
return isinstance(x, SymmetricFunctionAlgebra_generic)
|
def is_SymmetricFunctionAlgebra(x):
"\n Checks whether ``x`` is a symmetric function algebra.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import is_SymmetricFunctionAlgebra\n sage: is_SymmetricFunctionAlgebra(5)\n False\n sage: is_SymmetricFunctionAlgebra(ZZ)\n False\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(ZZ).schur())\n True\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).e())\n True\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).macdonald(q=1,t=1).P())\n True\n sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(FractionField(QQ['q','t'])).macdonald().P())\n True\n "
return isinstance(x, SymmetricFunctionAlgebra_generic)<|docstring|>Checks whether ``x`` is a symmetric function algebra.
EXAMPLES::
sage: from sage.combinat.sf.sfa import is_SymmetricFunctionAlgebra
sage: is_SymmetricFunctionAlgebra(5)
False
sage: is_SymmetricFunctionAlgebra(ZZ)
False
sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(ZZ).schur())
True
sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).e())
True
sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(QQ).macdonald(q=1,t=1).P())
True
sage: is_SymmetricFunctionAlgebra(SymmetricFunctions(FractionField(QQ['q','t'])).macdonald().P())
True<|endoftext|>
|
86ccddf9f598ed3b4ca77ac9963a5c33455e5283381610aa3baf277ac69c4595
|
def zee(part):
'\n Return the size of the centralizer of permutations of cycle type ``part``.\n\n Note that the size of the centralizer is the inner product between `p(part)` and\n itself where `p` is the power-sum symmetric functions.\n\n INPUT:\n\n - ``part`` -- an integer partition (for example, [2,1,1])\n\n OUTPUT:\n\n - the integer `\\prod_{i} i^{m_i(part)} m_i(part)!` where `m_i(part)` is\n the number of parts in the partition ``part`` equal to `i`\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import zee\n sage: zee([2,1,1])\n 4\n '
if (not isinstance(part, sage.combinat.partition.Partition)):
part = sage.combinat.partition.Partition(part)
return part.centralizer_size()
|
Return the size of the centralizer of permutations of cycle type ``part``.
Note that the size of the centralizer is the inner product between `p(part)` and
itself where `p` is the power-sum symmetric functions.
INPUT:
- ``part`` -- an integer partition (for example, [2,1,1])
OUTPUT:
- the integer `\prod_{i} i^{m_i(part)} m_i(part)!` where `m_i(part)` is
the number of parts in the partition ``part`` equal to `i`
EXAMPLES::
sage: from sage.combinat.sf.sfa import zee
sage: zee([2,1,1])
4
|
src/sage/combinat/sf/sfa.py
|
zee
|
bopopescu/sagesmc
| 5 |
python
|
def zee(part):
'\n Return the size of the centralizer of permutations of cycle type ``part``.\n\n Note that the size of the centralizer is the inner product between `p(part)` and\n itself where `p` is the power-sum symmetric functions.\n\n INPUT:\n\n - ``part`` -- an integer partition (for example, [2,1,1])\n\n OUTPUT:\n\n - the integer `\\prod_{i} i^{m_i(part)} m_i(part)!` where `m_i(part)` is\n the number of parts in the partition ``part`` equal to `i`\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import zee\n sage: zee([2,1,1])\n 4\n '
if (not isinstance(part, sage.combinat.partition.Partition)):
part = sage.combinat.partition.Partition(part)
return part.centralizer_size()
|
def zee(part):
'\n Return the size of the centralizer of permutations of cycle type ``part``.\n\n Note that the size of the centralizer is the inner product between `p(part)` and\n itself where `p` is the power-sum symmetric functions.\n\n INPUT:\n\n - ``part`` -- an integer partition (for example, [2,1,1])\n\n OUTPUT:\n\n - the integer `\\prod_{i} i^{m_i(part)} m_i(part)!` where `m_i(part)` is\n the number of parts in the partition ``part`` equal to `i`\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import zee\n sage: zee([2,1,1])\n 4\n '
if (not isinstance(part, sage.combinat.partition.Partition)):
part = sage.combinat.partition.Partition(part)
return part.centralizer_size()<|docstring|>Return the size of the centralizer of permutations of cycle type ``part``.
Note that the size of the centralizer is the inner product between `p(part)` and
itself where `p` is the power-sum symmetric functions.
INPUT:
- ``part`` -- an integer partition (for example, [2,1,1])
OUTPUT:
- the integer `\prod_{i} i^{m_i(part)} m_i(part)!` where `m_i(part)` is
the number of parts in the partition ``part`` equal to `i`
EXAMPLES::
sage: from sage.combinat.sf.sfa import zee
sage: zee([2,1,1])
4<|endoftext|>
|
0de71fb3fcca5d1022e4b1ca41b64b659015a54f8a95c3e084a9480f02f462b4
|
def is_SymmetricFunction(x):
'\n Checks whether ``x`` is a symmetric function.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import is_SymmetricFunction\n sage: s = SymmetricFunctions(QQ).s()\n sage: is_SymmetricFunction(2)\n False\n sage: is_SymmetricFunction(s(2))\n True\n sage: is_SymmetricFunction(s([2,1]))\n True\n '
return isinstance(x, SymmetricFunctionAlgebra_generic.Element)
|
Checks whether ``x`` is a symmetric function.
EXAMPLES::
sage: from sage.combinat.sf.sfa import is_SymmetricFunction
sage: s = SymmetricFunctions(QQ).s()
sage: is_SymmetricFunction(2)
False
sage: is_SymmetricFunction(s(2))
True
sage: is_SymmetricFunction(s([2,1]))
True
|
src/sage/combinat/sf/sfa.py
|
is_SymmetricFunction
|
bopopescu/sagesmc
| 5 |
python
|
def is_SymmetricFunction(x):
'\n Checks whether ``x`` is a symmetric function.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import is_SymmetricFunction\n sage: s = SymmetricFunctions(QQ).s()\n sage: is_SymmetricFunction(2)\n False\n sage: is_SymmetricFunction(s(2))\n True\n sage: is_SymmetricFunction(s([2,1]))\n True\n '
return isinstance(x, SymmetricFunctionAlgebra_generic.Element)
|
def is_SymmetricFunction(x):
'\n Checks whether ``x`` is a symmetric function.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import is_SymmetricFunction\n sage: s = SymmetricFunctions(QQ).s()\n sage: is_SymmetricFunction(2)\n False\n sage: is_SymmetricFunction(s(2))\n True\n sage: is_SymmetricFunction(s([2,1]))\n True\n '
return isinstance(x, SymmetricFunctionAlgebra_generic.Element)<|docstring|>Checks whether ``x`` is a symmetric function.
EXAMPLES::
sage: from sage.combinat.sf.sfa import is_SymmetricFunction
sage: s = SymmetricFunctions(QQ).s()
sage: is_SymmetricFunction(2)
False
sage: is_SymmetricFunction(s(2))
True
sage: is_SymmetricFunction(s([2,1]))
True<|endoftext|>
|
0502fa79c3f675efac898dab26d70ebbf7aaaa278efa25d73beb3ecc7e8134bb
|
def _lmax(x):
'\n Returns the max of ``x`` where ``x`` is a list.\n\n If ``x`` is the empty list, ``_lmax`` returns 0.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import _lmax\n sage: _lmax([3,2,1])\n 3\n sage: _lmax([])\n 0\n '
if (x == []):
return 0
else:
return max(x)
|
Returns the max of ``x`` where ``x`` is a list.
If ``x`` is the empty list, ``_lmax`` returns 0.
EXAMPLES::
sage: from sage.combinat.sf.sfa import _lmax
sage: _lmax([3,2,1])
3
sage: _lmax([])
0
|
src/sage/combinat/sf/sfa.py
|
_lmax
|
bopopescu/sagesmc
| 5 |
python
|
def _lmax(x):
'\n Returns the max of ``x`` where ``x`` is a list.\n\n If ``x`` is the empty list, ``_lmax`` returns 0.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import _lmax\n sage: _lmax([3,2,1])\n 3\n sage: _lmax([])\n 0\n '
if (x == []):
return 0
else:
return max(x)
|
def _lmax(x):
'\n Returns the max of ``x`` where ``x`` is a list.\n\n If ``x`` is the empty list, ``_lmax`` returns 0.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import _lmax\n sage: _lmax([3,2,1])\n 3\n sage: _lmax([])\n 0\n '
if (x == []):
return 0
else:
return max(x)<|docstring|>Returns the max of ``x`` where ``x`` is a list.
If ``x`` is the empty list, ``_lmax`` returns 0.
EXAMPLES::
sage: from sage.combinat.sf.sfa import _lmax
sage: _lmax([3,2,1])
3
sage: _lmax([])
0<|endoftext|>
|
8f9f08dfab5c77a80a6a6bb82da24f3b1b7be08fccb5cc4fe6789972ef345fa1
|
def _nonnegative_coefficients(x):
'\n Returns ``True`` if ``x`` has nonnegative coefficients.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import _nonnegative_coefficients\n sage: _nonnegative_coefficients(2)\n True\n sage: _nonnegative_coefficients(-2)\n False\n sage: R.<x> = ZZ[]\n sage: _nonnegative_coefficients(x^2+4)\n True\n sage: _nonnegative_coefficients(x^2-4)\n False\n '
if (is_Polynomial(x) or is_MPolynomial(x)):
return all([(c >= 0) for c in x.coeffs()])
else:
return (x >= 0)
|
Returns ``True`` if ``x`` has nonnegative coefficients.
EXAMPLES::
sage: from sage.combinat.sf.sfa import _nonnegative_coefficients
sage: _nonnegative_coefficients(2)
True
sage: _nonnegative_coefficients(-2)
False
sage: R.<x> = ZZ[]
sage: _nonnegative_coefficients(x^2+4)
True
sage: _nonnegative_coefficients(x^2-4)
False
|
src/sage/combinat/sf/sfa.py
|
_nonnegative_coefficients
|
bopopescu/sagesmc
| 5 |
python
|
def _nonnegative_coefficients(x):
'\n Returns ``True`` if ``x`` has nonnegative coefficients.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import _nonnegative_coefficients\n sage: _nonnegative_coefficients(2)\n True\n sage: _nonnegative_coefficients(-2)\n False\n sage: R.<x> = ZZ[]\n sage: _nonnegative_coefficients(x^2+4)\n True\n sage: _nonnegative_coefficients(x^2-4)\n False\n '
if (is_Polynomial(x) or is_MPolynomial(x)):
return all([(c >= 0) for c in x.coeffs()])
else:
return (x >= 0)
|
def _nonnegative_coefficients(x):
'\n Returns ``True`` if ``x`` has nonnegative coefficients.\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import _nonnegative_coefficients\n sage: _nonnegative_coefficients(2)\n True\n sage: _nonnegative_coefficients(-2)\n False\n sage: R.<x> = ZZ[]\n sage: _nonnegative_coefficients(x^2+4)\n True\n sage: _nonnegative_coefficients(x^2-4)\n False\n '
if (is_Polynomial(x) or is_MPolynomial(x)):
return all([(c >= 0) for c in x.coeffs()])
else:
return (x >= 0)<|docstring|>Returns ``True`` if ``x`` has nonnegative coefficients.
EXAMPLES::
sage: from sage.combinat.sf.sfa import _nonnegative_coefficients
sage: _nonnegative_coefficients(2)
True
sage: _nonnegative_coefficients(-2)
False
sage: R.<x> = ZZ[]
sage: _nonnegative_coefficients(x^2+4)
True
sage: _nonnegative_coefficients(x^2-4)
False<|endoftext|>
|
d6005f07df18344bd6631f1b85262bef6292cd49b32e598cfdae9be1d1b7a7a6
|
def __init__(self, base):
'\n Initialize the bases of the ring of symmetric functions.\n\n INPUT:\n\n - ``self`` -- a category of bases for the symmetric functions\n - ``base`` -- ring of symmetric functions\n\n TESTS::\n\n sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases\n sage: Sym = SymmetricFunctions(QQ)\n sage: bases = SymmetricFunctionsBases(Sym); bases\n Category of bases of Symmetric Functions over Rational Field\n sage: Sym.schur() in bases\n True\n '
Category_realization_of_parent.__init__(self, base)
|
Initialize the bases of the ring of symmetric functions.
INPUT:
- ``self`` -- a category of bases for the symmetric functions
- ``base`` -- ring of symmetric functions
TESTS::
sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases
sage: Sym = SymmetricFunctions(QQ)
sage: bases = SymmetricFunctionsBases(Sym); bases
Category of bases of Symmetric Functions over Rational Field
sage: Sym.schur() in bases
True
|
src/sage/combinat/sf/sfa.py
|
__init__
|
bopopescu/sagesmc
| 5 |
python
|
def __init__(self, base):
'\n Initialize the bases of the ring of symmetric functions.\n\n INPUT:\n\n - ``self`` -- a category of bases for the symmetric functions\n - ``base`` -- ring of symmetric functions\n\n TESTS::\n\n sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases\n sage: Sym = SymmetricFunctions(QQ)\n sage: bases = SymmetricFunctionsBases(Sym); bases\n Category of bases of Symmetric Functions over Rational Field\n sage: Sym.schur() in bases\n True\n '
Category_realization_of_parent.__init__(self, base)
|
def __init__(self, base):
'\n Initialize the bases of the ring of symmetric functions.\n\n INPUT:\n\n - ``self`` -- a category of bases for the symmetric functions\n - ``base`` -- ring of symmetric functions\n\n TESTS::\n\n sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases\n sage: Sym = SymmetricFunctions(QQ)\n sage: bases = SymmetricFunctionsBases(Sym); bases\n Category of bases of Symmetric Functions over Rational Field\n sage: Sym.schur() in bases\n True\n '
Category_realization_of_parent.__init__(self, base)<|docstring|>Initialize the bases of the ring of symmetric functions.
INPUT:
- ``self`` -- a category of bases for the symmetric functions
- ``base`` -- ring of symmetric functions
TESTS::
sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases
sage: Sym = SymmetricFunctions(QQ)
sage: bases = SymmetricFunctionsBases(Sym); bases
Category of bases of Symmetric Functions over Rational Field
sage: Sym.schur() in bases
True<|endoftext|>
|
3d65aa5b1b2f56eb50ccbee723d734d7972f7321d3ad85418514c582c316032b
|
def _repr_(self):
"\n Returns the representation of ``self``.\n\n INPUT:\n\n - ``self`` -- a category of bases for the symmetric functions\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases\n sage: Sym = SymmetricFunctions(QQ)\n sage: bases = SymmetricFunctionsBases(Sym)\n sage: bases._repr_()\n 'Category of bases of Symmetric Functions over Rational Field'\n "
return ('Category of bases of %s' % self.base())
|
Returns the representation of ``self``.
INPUT:
- ``self`` -- a category of bases for the symmetric functions
EXAMPLES::
sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases
sage: Sym = SymmetricFunctions(QQ)
sage: bases = SymmetricFunctionsBases(Sym)
sage: bases._repr_()
'Category of bases of Symmetric Functions over Rational Field'
|
src/sage/combinat/sf/sfa.py
|
_repr_
|
bopopescu/sagesmc
| 5 |
python
|
def _repr_(self):
"\n Returns the representation of ``self``.\n\n INPUT:\n\n - ``self`` -- a category of bases for the symmetric functions\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases\n sage: Sym = SymmetricFunctions(QQ)\n sage: bases = SymmetricFunctionsBases(Sym)\n sage: bases._repr_()\n 'Category of bases of Symmetric Functions over Rational Field'\n "
return ('Category of bases of %s' % self.base())
|
def _repr_(self):
"\n Returns the representation of ``self``.\n\n INPUT:\n\n - ``self`` -- a category of bases for the symmetric functions\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases\n sage: Sym = SymmetricFunctions(QQ)\n sage: bases = SymmetricFunctionsBases(Sym)\n sage: bases._repr_()\n 'Category of bases of Symmetric Functions over Rational Field'\n "
return ('Category of bases of %s' % self.base())<|docstring|>Returns the representation of ``self``.
INPUT:
- ``self`` -- a category of bases for the symmetric functions
EXAMPLES::
sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases
sage: Sym = SymmetricFunctions(QQ)
sage: bases = SymmetricFunctionsBases(Sym)
sage: bases._repr_()
'Category of bases of Symmetric Functions over Rational Field'<|endoftext|>
|
538828f3e5f7a9743ee37e30990396089405d1a76bed4ca419b8016c57cdd9e7
|
def super_categories(self):
'\n The super categories of ``self``.\n\n INPUT:\n\n - ``self`` -- a category of bases for the symmetric functions\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases\n sage: Sym = SymmetricFunctions(QQ)\n sage: bases = SymmetricFunctionsBases(Sym)\n sage: bases.super_categories()\n [Category of graded hopf algebras with basis over Rational Field, Category of realizations of Symmetric Functions over Rational Field, Category of commutative rings]\n '
from sage.categories.all import CommutativeRings, GradedHopfAlgebrasWithBasis
return [GradedHopfAlgebrasWithBasis(self.base().base_ring()), Realizations(self.base()), CommutativeRings()]
|
The super categories of ``self``.
INPUT:
- ``self`` -- a category of bases for the symmetric functions
EXAMPLES::
sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases
sage: Sym = SymmetricFunctions(QQ)
sage: bases = SymmetricFunctionsBases(Sym)
sage: bases.super_categories()
[Category of graded hopf algebras with basis over Rational Field, Category of realizations of Symmetric Functions over Rational Field, Category of commutative rings]
|
src/sage/combinat/sf/sfa.py
|
super_categories
|
bopopescu/sagesmc
| 5 |
python
|
def super_categories(self):
'\n The super categories of ``self``.\n\n INPUT:\n\n - ``self`` -- a category of bases for the symmetric functions\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases\n sage: Sym = SymmetricFunctions(QQ)\n sage: bases = SymmetricFunctionsBases(Sym)\n sage: bases.super_categories()\n [Category of graded hopf algebras with basis over Rational Field, Category of realizations of Symmetric Functions over Rational Field, Category of commutative rings]\n '
from sage.categories.all import CommutativeRings, GradedHopfAlgebrasWithBasis
return [GradedHopfAlgebrasWithBasis(self.base().base_ring()), Realizations(self.base()), CommutativeRings()]
|
def super_categories(self):
'\n The super categories of ``self``.\n\n INPUT:\n\n - ``self`` -- a category of bases for the symmetric functions\n\n EXAMPLES::\n\n sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases\n sage: Sym = SymmetricFunctions(QQ)\n sage: bases = SymmetricFunctionsBases(Sym)\n sage: bases.super_categories()\n [Category of graded hopf algebras with basis over Rational Field, Category of realizations of Symmetric Functions over Rational Field, Category of commutative rings]\n '
from sage.categories.all import CommutativeRings, GradedHopfAlgebrasWithBasis
return [GradedHopfAlgebrasWithBasis(self.base().base_ring()), Realizations(self.base()), CommutativeRings()]<|docstring|>The super categories of ``self``.
INPUT:
- ``self`` -- a category of bases for the symmetric functions
EXAMPLES::
sage: from sage.combinat.sf.sfa import SymmetricFunctionsBases
sage: Sym = SymmetricFunctions(QQ)
sage: bases = SymmetricFunctionsBases(Sym)
sage: bases.super_categories()
[Category of graded hopf algebras with basis over Rational Field, Category of realizations of Symmetric Functions over Rational Field, Category of commutative rings]<|endoftext|>
|
5bb87f55d68ff822272486e53ff716ba4072d9e06b044d723f5a48b6d459b9f6
|
def __init__(self, Sym, basis_name=None, prefix=None):
'\n Initializes the symmetric function algebra.\n\n INPUT:\n\n - ``Sym`` -- the ring of symmetric functions\n - ``basis_name`` -- name of basis (default: ``None``)\n - ``prefix`` -- prefix used to display basis\n\n TESTS::\n\n sage: from sage.combinat.sf.classical import SymmetricFunctionAlgebra_classical\n sage: s = SymmetricFunctions(QQ).s()\n sage: isinstance(s, SymmetricFunctionAlgebra_classical)\n True\n sage: TestSuite(s).run()\n '
R = Sym.base_ring()
from sage.categories.all import CommutativeRings
if (R not in CommutativeRings()):
raise TypeError('Argument R must be a commutative ring.')
try:
R(Integer(1))
except Exception:
raise ValueError('R must have a unit element')
if (basis_name is not None):
self._basis = basis_name
if (prefix is not None):
self._prefix = prefix
self._sym = Sym
CombinatorialFreeModule.__init__(self, Sym.base_ring(), _Partitions, category=SymmetricFunctionsBases(Sym), bracket='', prefix=prefix)
|
Initializes the symmetric function algebra.
INPUT:
- ``Sym`` -- the ring of symmetric functions
- ``basis_name`` -- name of basis (default: ``None``)
- ``prefix`` -- prefix used to display basis
TESTS::
sage: from sage.combinat.sf.classical import SymmetricFunctionAlgebra_classical
sage: s = SymmetricFunctions(QQ).s()
sage: isinstance(s, SymmetricFunctionAlgebra_classical)
True
sage: TestSuite(s).run()
|
src/sage/combinat/sf/sfa.py
|
__init__
|
bopopescu/sagesmc
| 5 |
python
|
def __init__(self, Sym, basis_name=None, prefix=None):
'\n Initializes the symmetric function algebra.\n\n INPUT:\n\n - ``Sym`` -- the ring of symmetric functions\n - ``basis_name`` -- name of basis (default: ``None``)\n - ``prefix`` -- prefix used to display basis\n\n TESTS::\n\n sage: from sage.combinat.sf.classical import SymmetricFunctionAlgebra_classical\n sage: s = SymmetricFunctions(QQ).s()\n sage: isinstance(s, SymmetricFunctionAlgebra_classical)\n True\n sage: TestSuite(s).run()\n '
R = Sym.base_ring()
from sage.categories.all import CommutativeRings
if (R not in CommutativeRings()):
raise TypeError('Argument R must be a commutative ring.')
try:
R(Integer(1))
except Exception:
raise ValueError('R must have a unit element')
if (basis_name is not None):
self._basis = basis_name
if (prefix is not None):
self._prefix = prefix
self._sym = Sym
CombinatorialFreeModule.__init__(self, Sym.base_ring(), _Partitions, category=SymmetricFunctionsBases(Sym), bracket=, prefix=prefix)
|
def __init__(self, Sym, basis_name=None, prefix=None):
'\n Initializes the symmetric function algebra.\n\n INPUT:\n\n - ``Sym`` -- the ring of symmetric functions\n - ``basis_name`` -- name of basis (default: ``None``)\n - ``prefix`` -- prefix used to display basis\n\n TESTS::\n\n sage: from sage.combinat.sf.classical import SymmetricFunctionAlgebra_classical\n sage: s = SymmetricFunctions(QQ).s()\n sage: isinstance(s, SymmetricFunctionAlgebra_classical)\n True\n sage: TestSuite(s).run()\n '
R = Sym.base_ring()
from sage.categories.all import CommutativeRings
if (R not in CommutativeRings()):
raise TypeError('Argument R must be a commutative ring.')
try:
R(Integer(1))
except Exception:
raise ValueError('R must have a unit element')
if (basis_name is not None):
self._basis = basis_name
if (prefix is not None):
self._prefix = prefix
self._sym = Sym
CombinatorialFreeModule.__init__(self, Sym.base_ring(), _Partitions, category=SymmetricFunctionsBases(Sym), bracket=, prefix=prefix)<|docstring|>Initializes the symmetric function algebra.
INPUT:
- ``Sym`` -- the ring of symmetric functions
- ``basis_name`` -- name of basis (default: ``None``)
- ``prefix`` -- prefix used to display basis
TESTS::
sage: from sage.combinat.sf.classical import SymmetricFunctionAlgebra_classical
sage: s = SymmetricFunctions(QQ).s()
sage: isinstance(s, SymmetricFunctionAlgebra_classical)
True
sage: TestSuite(s).run()<|endoftext|>
|
01f61ddc7e56331fc506f3ab13caf8b9bf0dabfcb398ccf9df932e6e71fa350f
|
def __getitem__(self, c, *rest):
"\n This method implements the abuses of notations ``p[2,1]``,\n ``p[[2,1]]``, ``p[Partition([2,1])]``.\n\n INPUT:\n\n - ``c`` -- a list, list of lists, or partition\n\n .. TODO::\n\n Should call ``super.term`` so as not to interfere with the\n standard notation ``p['x,y,z']``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s[2,1]\n s[2, 1]\n sage: s[[2,1]]\n s[2, 1]\n sage: s[Partition([2,1])]\n s[2, 1]\n "
C = self.basis().keys()
if isinstance(c, C.element_class):
if (len(rest) != 0):
raise ValueError('invalid number of arguments')
elif ((len(rest) > 0) or (type(c) is int) or (type(c) is Integer)):
c = C(([c] + list(rest)))
else:
c = C(list(c))
return self.monomial(c)
|
This method implements the abuses of notations ``p[2,1]``,
``p[[2,1]]``, ``p[Partition([2,1])]``.
INPUT:
- ``c`` -- a list, list of lists, or partition
.. TODO::
Should call ``super.term`` so as not to interfere with the
standard notation ``p['x,y,z']``.
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: s[2,1]
s[2, 1]
sage: s[[2,1]]
s[2, 1]
sage: s[Partition([2,1])]
s[2, 1]
|
src/sage/combinat/sf/sfa.py
|
__getitem__
|
bopopescu/sagesmc
| 5 |
python
|
def __getitem__(self, c, *rest):
"\n This method implements the abuses of notations ``p[2,1]``,\n ``p[[2,1]]``, ``p[Partition([2,1])]``.\n\n INPUT:\n\n - ``c`` -- a list, list of lists, or partition\n\n .. TODO::\n\n Should call ``super.term`` so as not to interfere with the\n standard notation ``p['x,y,z']``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s[2,1]\n s[2, 1]\n sage: s[[2,1]]\n s[2, 1]\n sage: s[Partition([2,1])]\n s[2, 1]\n "
C = self.basis().keys()
if isinstance(c, C.element_class):
if (len(rest) != 0):
raise ValueError('invalid number of arguments')
elif ((len(rest) > 0) or (type(c) is int) or (type(c) is Integer)):
c = C(([c] + list(rest)))
else:
c = C(list(c))
return self.monomial(c)
|
def __getitem__(self, c, *rest):
"\n This method implements the abuses of notations ``p[2,1]``,\n ``p[[2,1]]``, ``p[Partition([2,1])]``.\n\n INPUT:\n\n - ``c`` -- a list, list of lists, or partition\n\n .. TODO::\n\n Should call ``super.term`` so as not to interfere with the\n standard notation ``p['x,y,z']``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s[2,1]\n s[2, 1]\n sage: s[[2,1]]\n s[2, 1]\n sage: s[Partition([2,1])]\n s[2, 1]\n "
C = self.basis().keys()
if isinstance(c, C.element_class):
if (len(rest) != 0):
raise ValueError('invalid number of arguments')
elif ((len(rest) > 0) or (type(c) is int) or (type(c) is Integer)):
c = C(([c] + list(rest)))
else:
c = C(list(c))
return self.monomial(c)<|docstring|>This method implements the abuses of notations ``p[2,1]``,
``p[[2,1]]``, ``p[Partition([2,1])]``.
INPUT:
- ``c`` -- a list, list of lists, or partition
.. TODO::
Should call ``super.term`` so as not to interfere with the
standard notation ``p['x,y,z']``.
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: s[2,1]
s[2, 1]
sage: s[[2,1]]
s[2, 1]
sage: s[Partition([2,1])]
s[2, 1]<|endoftext|>
|
6948a0a518543fd3a021dab16bf8e3769405b341819ef6d142e2be5093d1a6cb
|
def _change_by_proportionality(self, x, function):
'\n Return the symmetric function obtained from ``x`` by scaling\n each basis element corresponding to the partition `\\lambda` by\n ``function``(`\\lambda`).\n\n INPUT:\n\n - ``x`` -- a symmetric function\n - ``function`` -- a function which takes in a partition\n and returns a scalar\n\n OUTPUT:\n\n A symmetric function in ``self`` which is a scaled version of ``x``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([3])+s([2,1])+s([1,1,1]); a\n s[1, 1, 1] + s[2, 1] + s[3]\n sage: f = lambda part: len(part)\n sage: s._change_by_proportionality(a, f)\n 3*s[1, 1, 1] + 2*s[2, 1] + s[3]\n '
BR = self.base_ring()
z_elt = {}
for (m, c) in x._monomial_coefficients.iteritems():
coeff = function(m)
z_elt[m] = BR((c * coeff))
return self._from_dict(z_elt)
|
Return the symmetric function obtained from ``x`` by scaling
each basis element corresponding to the partition `\lambda` by
``function``(`\lambda`).
INPUT:
- ``x`` -- a symmetric function
- ``function`` -- a function which takes in a partition
and returns a scalar
OUTPUT:
A symmetric function in ``self`` which is a scaled version of ``x``.
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([3])+s([2,1])+s([1,1,1]); a
s[1, 1, 1] + s[2, 1] + s[3]
sage: f = lambda part: len(part)
sage: s._change_by_proportionality(a, f)
3*s[1, 1, 1] + 2*s[2, 1] + s[3]
|
src/sage/combinat/sf/sfa.py
|
_change_by_proportionality
|
bopopescu/sagesmc
| 5 |
python
|
def _change_by_proportionality(self, x, function):
'\n Return the symmetric function obtained from ``x`` by scaling\n each basis element corresponding to the partition `\\lambda` by\n ``function``(`\\lambda`).\n\n INPUT:\n\n - ``x`` -- a symmetric function\n - ``function`` -- a function which takes in a partition\n and returns a scalar\n\n OUTPUT:\n\n A symmetric function in ``self`` which is a scaled version of ``x``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([3])+s([2,1])+s([1,1,1]); a\n s[1, 1, 1] + s[2, 1] + s[3]\n sage: f = lambda part: len(part)\n sage: s._change_by_proportionality(a, f)\n 3*s[1, 1, 1] + 2*s[2, 1] + s[3]\n '
BR = self.base_ring()
z_elt = {}
for (m, c) in x._monomial_coefficients.iteritems():
coeff = function(m)
z_elt[m] = BR((c * coeff))
return self._from_dict(z_elt)
|
def _change_by_proportionality(self, x, function):
'\n Return the symmetric function obtained from ``x`` by scaling\n each basis element corresponding to the partition `\\lambda` by\n ``function``(`\\lambda`).\n\n INPUT:\n\n - ``x`` -- a symmetric function\n - ``function`` -- a function which takes in a partition\n and returns a scalar\n\n OUTPUT:\n\n A symmetric function in ``self`` which is a scaled version of ``x``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([3])+s([2,1])+s([1,1,1]); a\n s[1, 1, 1] + s[2, 1] + s[3]\n sage: f = lambda part: len(part)\n sage: s._change_by_proportionality(a, f)\n 3*s[1, 1, 1] + 2*s[2, 1] + s[3]\n '
BR = self.base_ring()
z_elt = {}
for (m, c) in x._monomial_coefficients.iteritems():
coeff = function(m)
z_elt[m] = BR((c * coeff))
return self._from_dict(z_elt)<|docstring|>Return the symmetric function obtained from ``x`` by scaling
each basis element corresponding to the partition `\lambda` by
``function``(`\lambda`).
INPUT:
- ``x`` -- a symmetric function
- ``function`` -- a function which takes in a partition
and returns a scalar
OUTPUT:
A symmetric function in ``self`` which is a scaled version of ``x``.
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([3])+s([2,1])+s([1,1,1]); a
s[1, 1, 1] + s[2, 1] + s[3]
sage: f = lambda part: len(part)
sage: s._change_by_proportionality(a, f)
3*s[1, 1, 1] + 2*s[2, 1] + s[3]<|endoftext|>
|
ee2ad53d81c1d99cf0dab4bbb5e3db754b653f84fffbb17bbddf1c9ff812f0a5
|
def _change_by_plethysm(self, x, expr, deg_one):
'\n Return the plethysm of ``x`` by ``expr``.\n\n INPUT:\n\n - ``x` -- a symmetric function\n - ``expr`` -- an expression used in the plethysm\n - ``deg_one`` -- a list (or iterable) specifying the degree one\n variables (that is, the terms to be treated as degree-one\n elements when encountered in ``x``; they will be taken to the\n appropriate powers when computing the plethysm)\n\n OUTPUT:\n\n The plethysm of ``x`` by ``expr``.\n\n EXAMPLES::\n\n sage: m = SymmetricFunctions(QQ).m()\n sage: a = m([2,1])\n sage: a.omega()\n -m[2, 1] - 2*m[3]\n sage: m._change_by_plethysm(-a,-1,[])\n -m[2, 1] - 2*m[3]\n\n ::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([3])\n sage: s._change_by_plethysm(-a,-1,[])\n s[1, 1, 1]\n '
p = self.realization_of().power()
p_x = p(x)
expr_k = (lambda k: expr.subs(**dict([(str(x), (x ** k)) for x in deg_one])))
f = (lambda m, c: (m, (c * prod([expr_k(k) for k in m]))))
return self(p_x.map_item(f))
|
Return the plethysm of ``x`` by ``expr``.
INPUT:
- ``x` -- a symmetric function
- ``expr`` -- an expression used in the plethysm
- ``deg_one`` -- a list (or iterable) specifying the degree one
variables (that is, the terms to be treated as degree-one
elements when encountered in ``x``; they will be taken to the
appropriate powers when computing the plethysm)
OUTPUT:
The plethysm of ``x`` by ``expr``.
EXAMPLES::
sage: m = SymmetricFunctions(QQ).m()
sage: a = m([2,1])
sage: a.omega()
-m[2, 1] - 2*m[3]
sage: m._change_by_plethysm(-a,-1,[])
-m[2, 1] - 2*m[3]
::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([3])
sage: s._change_by_plethysm(-a,-1,[])
s[1, 1, 1]
|
src/sage/combinat/sf/sfa.py
|
_change_by_plethysm
|
bopopescu/sagesmc
| 5 |
python
|
def _change_by_plethysm(self, x, expr, deg_one):
'\n Return the plethysm of ``x`` by ``expr``.\n\n INPUT:\n\n - ``x` -- a symmetric function\n - ``expr`` -- an expression used in the plethysm\n - ``deg_one`` -- a list (or iterable) specifying the degree one\n variables (that is, the terms to be treated as degree-one\n elements when encountered in ``x``; they will be taken to the\n appropriate powers when computing the plethysm)\n\n OUTPUT:\n\n The plethysm of ``x`` by ``expr``.\n\n EXAMPLES::\n\n sage: m = SymmetricFunctions(QQ).m()\n sage: a = m([2,1])\n sage: a.omega()\n -m[2, 1] - 2*m[3]\n sage: m._change_by_plethysm(-a,-1,[])\n -m[2, 1] - 2*m[3]\n\n ::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([3])\n sage: s._change_by_plethysm(-a,-1,[])\n s[1, 1, 1]\n '
p = self.realization_of().power()
p_x = p(x)
expr_k = (lambda k: expr.subs(**dict([(str(x), (x ** k)) for x in deg_one])))
f = (lambda m, c: (m, (c * prod([expr_k(k) for k in m]))))
return self(p_x.map_item(f))
|
def _change_by_plethysm(self, x, expr, deg_one):
'\n Return the plethysm of ``x`` by ``expr``.\n\n INPUT:\n\n - ``x` -- a symmetric function\n - ``expr`` -- an expression used in the plethysm\n - ``deg_one`` -- a list (or iterable) specifying the degree one\n variables (that is, the terms to be treated as degree-one\n elements when encountered in ``x``; they will be taken to the\n appropriate powers when computing the plethysm)\n\n OUTPUT:\n\n The plethysm of ``x`` by ``expr``.\n\n EXAMPLES::\n\n sage: m = SymmetricFunctions(QQ).m()\n sage: a = m([2,1])\n sage: a.omega()\n -m[2, 1] - 2*m[3]\n sage: m._change_by_plethysm(-a,-1,[])\n -m[2, 1] - 2*m[3]\n\n ::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([3])\n sage: s._change_by_plethysm(-a,-1,[])\n s[1, 1, 1]\n '
p = self.realization_of().power()
p_x = p(x)
expr_k = (lambda k: expr.subs(**dict([(str(x), (x ** k)) for x in deg_one])))
f = (lambda m, c: (m, (c * prod([expr_k(k) for k in m]))))
return self(p_x.map_item(f))<|docstring|>Return the plethysm of ``x`` by ``expr``.
INPUT:
- ``x` -- a symmetric function
- ``expr`` -- an expression used in the plethysm
- ``deg_one`` -- a list (or iterable) specifying the degree one
variables (that is, the terms to be treated as degree-one
elements when encountered in ``x``; they will be taken to the
appropriate powers when computing the plethysm)
OUTPUT:
The plethysm of ``x`` by ``expr``.
EXAMPLES::
sage: m = SymmetricFunctions(QQ).m()
sage: a = m([2,1])
sage: a.omega()
-m[2, 1] - 2*m[3]
sage: m._change_by_plethysm(-a,-1,[])
-m[2, 1] - 2*m[3]
::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([3])
sage: s._change_by_plethysm(-a,-1,[])
s[1, 1, 1]<|endoftext|>
|
fdedded23af43049121ea168f04d135632824b62af3e7ba6e2e8e3b9f5564e66
|
def _apply_multi_module_morphism(self, x, y, f, orthogonal=False):
'\n Applies morphism specified by ``f`` on (``x``,``y``).\n\n INPUT:\n\n - ``x`` -- an element of ``self``\n - ``y`` -- an element of ``self``\n - ``f`` -- a function that takes in two partitions\n (basis elements) and returns an element of the target domain\n - ``orthogonal`` -- if orthogonal is set to ``True``, then\n ``f(part1, part2)`` is assumed to be 0 if ``part1 != part2``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([2,1])+s([1,1,1])\n sage: b = s([3])+s([2,1])\n sage: f1 = lambda p1, p2: len(p1)*len(p2)\n sage: f2 = lambda p1, p2: len(p1)+len(p2)\n sage: s._apply_multi_module_morphism(a,b,f1,orthogonal=False) #(2+3)*(2+1)\n 15\n sage: s._apply_multi_module_morphism(a,b,f1,orthogonal=True) #(2)*(2)\n 4\n sage: s._apply_multi_module_morphism(a,b,f2,orthogonal=False) #2*(2+3+2+1)\n 16\n sage: s._apply_multi_module_morphism(a,b,f2,orthogonal=True) #2+2\n 4\n '
res = 0
if orthogonal:
for (mx, cx) in x._monomial_coefficients.iteritems():
if (mx not in y._monomial_coefficients):
continue
else:
cy = y._monomial_coefficients[mx]
res += ((cx * cy) * f(mx, mx))
return res
else:
for (mx, cx) in x._monomial_coefficients.iteritems():
for (my, cy) in y._monomial_coefficients.iteritems():
res += ((cx * cy) * f(mx, my))
return res
|
Applies morphism specified by ``f`` on (``x``,``y``).
INPUT:
- ``x`` -- an element of ``self``
- ``y`` -- an element of ``self``
- ``f`` -- a function that takes in two partitions
(basis elements) and returns an element of the target domain
- ``orthogonal`` -- if orthogonal is set to ``True``, then
``f(part1, part2)`` is assumed to be 0 if ``part1 != part2``.
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([2,1])+s([1,1,1])
sage: b = s([3])+s([2,1])
sage: f1 = lambda p1, p2: len(p1)*len(p2)
sage: f2 = lambda p1, p2: len(p1)+len(p2)
sage: s._apply_multi_module_morphism(a,b,f1,orthogonal=False) #(2+3)*(2+1)
15
sage: s._apply_multi_module_morphism(a,b,f1,orthogonal=True) #(2)*(2)
4
sage: s._apply_multi_module_morphism(a,b,f2,orthogonal=False) #2*(2+3+2+1)
16
sage: s._apply_multi_module_morphism(a,b,f2,orthogonal=True) #2+2
4
|
src/sage/combinat/sf/sfa.py
|
_apply_multi_module_morphism
|
bopopescu/sagesmc
| 5 |
python
|
def _apply_multi_module_morphism(self, x, y, f, orthogonal=False):
'\n Applies morphism specified by ``f`` on (``x``,``y``).\n\n INPUT:\n\n - ``x`` -- an element of ``self``\n - ``y`` -- an element of ``self``\n - ``f`` -- a function that takes in two partitions\n (basis elements) and returns an element of the target domain\n - ``orthogonal`` -- if orthogonal is set to ``True``, then\n ``f(part1, part2)`` is assumed to be 0 if ``part1 != part2``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([2,1])+s([1,1,1])\n sage: b = s([3])+s([2,1])\n sage: f1 = lambda p1, p2: len(p1)*len(p2)\n sage: f2 = lambda p1, p2: len(p1)+len(p2)\n sage: s._apply_multi_module_morphism(a,b,f1,orthogonal=False) #(2+3)*(2+1)\n 15\n sage: s._apply_multi_module_morphism(a,b,f1,orthogonal=True) #(2)*(2)\n 4\n sage: s._apply_multi_module_morphism(a,b,f2,orthogonal=False) #2*(2+3+2+1)\n 16\n sage: s._apply_multi_module_morphism(a,b,f2,orthogonal=True) #2+2\n 4\n '
res = 0
if orthogonal:
for (mx, cx) in x._monomial_coefficients.iteritems():
if (mx not in y._monomial_coefficients):
continue
else:
cy = y._monomial_coefficients[mx]
res += ((cx * cy) * f(mx, mx))
return res
else:
for (mx, cx) in x._monomial_coefficients.iteritems():
for (my, cy) in y._monomial_coefficients.iteritems():
res += ((cx * cy) * f(mx, my))
return res
|
def _apply_multi_module_morphism(self, x, y, f, orthogonal=False):
'\n Applies morphism specified by ``f`` on (``x``,``y``).\n\n INPUT:\n\n - ``x`` -- an element of ``self``\n - ``y`` -- an element of ``self``\n - ``f`` -- a function that takes in two partitions\n (basis elements) and returns an element of the target domain\n - ``orthogonal`` -- if orthogonal is set to ``True``, then\n ``f(part1, part2)`` is assumed to be 0 if ``part1 != part2``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([2,1])+s([1,1,1])\n sage: b = s([3])+s([2,1])\n sage: f1 = lambda p1, p2: len(p1)*len(p2)\n sage: f2 = lambda p1, p2: len(p1)+len(p2)\n sage: s._apply_multi_module_morphism(a,b,f1,orthogonal=False) #(2+3)*(2+1)\n 15\n sage: s._apply_multi_module_morphism(a,b,f1,orthogonal=True) #(2)*(2)\n 4\n sage: s._apply_multi_module_morphism(a,b,f2,orthogonal=False) #2*(2+3+2+1)\n 16\n sage: s._apply_multi_module_morphism(a,b,f2,orthogonal=True) #2+2\n 4\n '
res = 0
if orthogonal:
for (mx, cx) in x._monomial_coefficients.iteritems():
if (mx not in y._monomial_coefficients):
continue
else:
cy = y._monomial_coefficients[mx]
res += ((cx * cy) * f(mx, mx))
return res
else:
for (mx, cx) in x._monomial_coefficients.iteritems():
for (my, cy) in y._monomial_coefficients.iteritems():
res += ((cx * cy) * f(mx, my))
return res<|docstring|>Applies morphism specified by ``f`` on (``x``,``y``).
INPUT:
- ``x`` -- an element of ``self``
- ``y`` -- an element of ``self``
- ``f`` -- a function that takes in two partitions
(basis elements) and returns an element of the target domain
- ``orthogonal`` -- if orthogonal is set to ``True``, then
``f(part1, part2)`` is assumed to be 0 if ``part1 != part2``.
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([2,1])+s([1,1,1])
sage: b = s([3])+s([2,1])
sage: f1 = lambda p1, p2: len(p1)*len(p2)
sage: f2 = lambda p1, p2: len(p1)+len(p2)
sage: s._apply_multi_module_morphism(a,b,f1,orthogonal=False) #(2+3)*(2+1)
15
sage: s._apply_multi_module_morphism(a,b,f1,orthogonal=True) #(2)*(2)
4
sage: s._apply_multi_module_morphism(a,b,f2,orthogonal=False) #2*(2+3+2+1)
16
sage: s._apply_multi_module_morphism(a,b,f2,orthogonal=True) #2+2
4<|endoftext|>
|
a617703b0ddf560fd5131add55080878f9d556c7c79add943cb2e902194105a8
|
def _from_element(self, x):
"\n Return the element of ``self`` with the same 'internal structure' as\n ``x``. This means the element whose coefficients in the basis ``self``\n are the respective coefficients of ``x`` in the basis of ``x``.\n\n INPUT:\n\n - ``x`` -- a symmetric function\n\n EXAMPLES::\n\n sage: e = SymmetricFunctions(QQ).e()\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = e([2,1]) + e([1,1,1]); a\n e[1, 1, 1] + e[2, 1]\n sage: s._from_element(a)\n s[1, 1, 1] + s[2, 1]\n "
return self._from_dict(x.monomial_coefficients())
|
Return the element of ``self`` with the same 'internal structure' as
``x``. This means the element whose coefficients in the basis ``self``
are the respective coefficients of ``x`` in the basis of ``x``.
INPUT:
- ``x`` -- a symmetric function
EXAMPLES::
sage: e = SymmetricFunctions(QQ).e()
sage: s = SymmetricFunctions(QQ).s()
sage: a = e([2,1]) + e([1,1,1]); a
e[1, 1, 1] + e[2, 1]
sage: s._from_element(a)
s[1, 1, 1] + s[2, 1]
|
src/sage/combinat/sf/sfa.py
|
_from_element
|
bopopescu/sagesmc
| 5 |
python
|
def _from_element(self, x):
"\n Return the element of ``self`` with the same 'internal structure' as\n ``x``. This means the element whose coefficients in the basis ``self``\n are the respective coefficients of ``x`` in the basis of ``x``.\n\n INPUT:\n\n - ``x`` -- a symmetric function\n\n EXAMPLES::\n\n sage: e = SymmetricFunctions(QQ).e()\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = e([2,1]) + e([1,1,1]); a\n e[1, 1, 1] + e[2, 1]\n sage: s._from_element(a)\n s[1, 1, 1] + s[2, 1]\n "
return self._from_dict(x.monomial_coefficients())
|
def _from_element(self, x):
"\n Return the element of ``self`` with the same 'internal structure' as\n ``x``. This means the element whose coefficients in the basis ``self``\n are the respective coefficients of ``x`` in the basis of ``x``.\n\n INPUT:\n\n - ``x`` -- a symmetric function\n\n EXAMPLES::\n\n sage: e = SymmetricFunctions(QQ).e()\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = e([2,1]) + e([1,1,1]); a\n e[1, 1, 1] + e[2, 1]\n sage: s._from_element(a)\n s[1, 1, 1] + s[2, 1]\n "
return self._from_dict(x.monomial_coefficients())<|docstring|>Return the element of ``self`` with the same 'internal structure' as
``x``. This means the element whose coefficients in the basis ``self``
are the respective coefficients of ``x`` in the basis of ``x``.
INPUT:
- ``x`` -- a symmetric function
EXAMPLES::
sage: e = SymmetricFunctions(QQ).e()
sage: s = SymmetricFunctions(QQ).s()
sage: a = e([2,1]) + e([1,1,1]); a
e[1, 1, 1] + e[2, 1]
sage: s._from_element(a)
s[1, 1, 1] + s[2, 1]<|endoftext|>
|
3d12551bf7937fe717b80d690485e696b0156342b8b03860961db8854fed2778
|
def _from_cache(self, element, cache_function, cache_dict, **subs_dict):
'\n Return the image of an element ``element`` of some realization `M`\n of the ring of symmetric functions under a linear map from `M` to\n ``self`` whose matrix representation is cached (with ``cache_dict``\n serving as cache, and ``cache_function`` as the function which\n precomputes this cache).\n\n INPUT:\n\n - ``element`` -- an element of a realization `M` of the ring of\n symmetric functions. Note that `M` can be a different realization\n than the one in which ``self`` is written, and does not have to\n be specified. It is assumed that the basis of ``self`` is indexed\n by partitions, and the degree of a basis element is the size of\n the partition indexing it.\n\n - ``cache_function`` -- a function which accepts an\n integer `n` as its input and creates the cache for that homogeneous\n component (saving it in ``cache_dict``).\n\n - ``cache_dict`` -- a dictionary storing a cache.\n It should be indexed by the positive integers `n`. Its values\n are dictionaries indexed by the partitions of size `n`. The values\n of those latter dictionaries are, again, dictionaries indexed by\n partitions of size `n`. Altogether, ``cache_dict`` should be\n understood to encode a graded linear map from `M` to the\n realization ``self`` of the ring of symmetric functions; the\n encoding is done in such a way that, for any `n` and any partitions\n ``lam`` and ``mu`` of `n`, the ``self[mu]``-coordinate of the image\n of ``M[lam]`` under this linear map (in the basis ``self``) is\n ``cache_dict[lam][mu]``.\n\n - ``subs_dict`` -- (optional) a dictionary for any substitutions\n to make after the value is extracted from ``cache_dict``.\n\n EXAMPLES::\n\n sage: R.<x> = QQ[]\n sage: Sym = SymmetricFunctions(R)\n sage: s = Sym.s()\n sage: p21 = Partition([2,1])\n sage: a = s(p21)\n sage: e = Sym.e()\n sage: cache_dict = {}\n sage: cache_dict[3] = {}\n sage: cache_dict[3][p21] = {}\n sage: cache_dict[3][p21][p21] = x^2\n sage: cache_dict[3][p21][Partition([1,1,1])] = 3*x\n sage: cache_function = lambda n: 0 #do nothing\n sage: e._from_cache(a, cache_function, cache_dict)\n 3*x*e[1, 1, 1] + x^2*e[2, 1]\n sage: e._from_cache(a, cache_function, cache_dict, x=2)\n 6*e[1, 1, 1] + 4*e[2, 1]\n '
BR = self.base_ring()
zero = BR.zero()
z_elt = {}
for (part, c) in element.monomial_coefficients().iteritems():
if (sum(part) not in cache_dict):
cache_function(sum(part))
part = _Partitions(part)
for (part2, c2) in cache_dict[sum(part)][part].iteritems():
if hasattr(c2, 'subs'):
c3 = (c * BR(c2.subs(**subs_dict)))
else:
c3 = (c * BR(c2))
z_elt[part2] = (z_elt.get(part2, zero) + BR(c3))
return self._from_dict(z_elt)
|
Return the image of an element ``element`` of some realization `M`
of the ring of symmetric functions under a linear map from `M` to
``self`` whose matrix representation is cached (with ``cache_dict``
serving as cache, and ``cache_function`` as the function which
precomputes this cache).
INPUT:
- ``element`` -- an element of a realization `M` of the ring of
symmetric functions. Note that `M` can be a different realization
than the one in which ``self`` is written, and does not have to
be specified. It is assumed that the basis of ``self`` is indexed
by partitions, and the degree of a basis element is the size of
the partition indexing it.
- ``cache_function`` -- a function which accepts an
integer `n` as its input and creates the cache for that homogeneous
component (saving it in ``cache_dict``).
- ``cache_dict`` -- a dictionary storing a cache.
It should be indexed by the positive integers `n`. Its values
are dictionaries indexed by the partitions of size `n`. The values
of those latter dictionaries are, again, dictionaries indexed by
partitions of size `n`. Altogether, ``cache_dict`` should be
understood to encode a graded linear map from `M` to the
realization ``self`` of the ring of symmetric functions; the
encoding is done in such a way that, for any `n` and any partitions
``lam`` and ``mu`` of `n`, the ``self[mu]``-coordinate of the image
of ``M[lam]`` under this linear map (in the basis ``self``) is
``cache_dict[lam][mu]``.
- ``subs_dict`` -- (optional) a dictionary for any substitutions
to make after the value is extracted from ``cache_dict``.
EXAMPLES::
sage: R.<x> = QQ[]
sage: Sym = SymmetricFunctions(R)
sage: s = Sym.s()
sage: p21 = Partition([2,1])
sage: a = s(p21)
sage: e = Sym.e()
sage: cache_dict = {}
sage: cache_dict[3] = {}
sage: cache_dict[3][p21] = {}
sage: cache_dict[3][p21][p21] = x^2
sage: cache_dict[3][p21][Partition([1,1,1])] = 3*x
sage: cache_function = lambda n: 0 #do nothing
sage: e._from_cache(a, cache_function, cache_dict)
3*x*e[1, 1, 1] + x^2*e[2, 1]
sage: e._from_cache(a, cache_function, cache_dict, x=2)
6*e[1, 1, 1] + 4*e[2, 1]
|
src/sage/combinat/sf/sfa.py
|
_from_cache
|
bopopescu/sagesmc
| 5 |
python
|
def _from_cache(self, element, cache_function, cache_dict, **subs_dict):
'\n Return the image of an element ``element`` of some realization `M`\n of the ring of symmetric functions under a linear map from `M` to\n ``self`` whose matrix representation is cached (with ``cache_dict``\n serving as cache, and ``cache_function`` as the function which\n precomputes this cache).\n\n INPUT:\n\n - ``element`` -- an element of a realization `M` of the ring of\n symmetric functions. Note that `M` can be a different realization\n than the one in which ``self`` is written, and does not have to\n be specified. It is assumed that the basis of ``self`` is indexed\n by partitions, and the degree of a basis element is the size of\n the partition indexing it.\n\n - ``cache_function`` -- a function which accepts an\n integer `n` as its input and creates the cache for that homogeneous\n component (saving it in ``cache_dict``).\n\n - ``cache_dict`` -- a dictionary storing a cache.\n It should be indexed by the positive integers `n`. Its values\n are dictionaries indexed by the partitions of size `n`. The values\n of those latter dictionaries are, again, dictionaries indexed by\n partitions of size `n`. Altogether, ``cache_dict`` should be\n understood to encode a graded linear map from `M` to the\n realization ``self`` of the ring of symmetric functions; the\n encoding is done in such a way that, for any `n` and any partitions\n ``lam`` and ``mu`` of `n`, the ``self[mu]``-coordinate of the image\n of ``M[lam]`` under this linear map (in the basis ``self``) is\n ``cache_dict[lam][mu]``.\n\n - ``subs_dict`` -- (optional) a dictionary for any substitutions\n to make after the value is extracted from ``cache_dict``.\n\n EXAMPLES::\n\n sage: R.<x> = QQ[]\n sage: Sym = SymmetricFunctions(R)\n sage: s = Sym.s()\n sage: p21 = Partition([2,1])\n sage: a = s(p21)\n sage: e = Sym.e()\n sage: cache_dict = {}\n sage: cache_dict[3] = {}\n sage: cache_dict[3][p21] = {}\n sage: cache_dict[3][p21][p21] = x^2\n sage: cache_dict[3][p21][Partition([1,1,1])] = 3*x\n sage: cache_function = lambda n: 0 #do nothing\n sage: e._from_cache(a, cache_function, cache_dict)\n 3*x*e[1, 1, 1] + x^2*e[2, 1]\n sage: e._from_cache(a, cache_function, cache_dict, x=2)\n 6*e[1, 1, 1] + 4*e[2, 1]\n '
BR = self.base_ring()
zero = BR.zero()
z_elt = {}
for (part, c) in element.monomial_coefficients().iteritems():
if (sum(part) not in cache_dict):
cache_function(sum(part))
part = _Partitions(part)
for (part2, c2) in cache_dict[sum(part)][part].iteritems():
if hasattr(c2, 'subs'):
c3 = (c * BR(c2.subs(**subs_dict)))
else:
c3 = (c * BR(c2))
z_elt[part2] = (z_elt.get(part2, zero) + BR(c3))
return self._from_dict(z_elt)
|
def _from_cache(self, element, cache_function, cache_dict, **subs_dict):
'\n Return the image of an element ``element`` of some realization `M`\n of the ring of symmetric functions under a linear map from `M` to\n ``self`` whose matrix representation is cached (with ``cache_dict``\n serving as cache, and ``cache_function`` as the function which\n precomputes this cache).\n\n INPUT:\n\n - ``element`` -- an element of a realization `M` of the ring of\n symmetric functions. Note that `M` can be a different realization\n than the one in which ``self`` is written, and does not have to\n be specified. It is assumed that the basis of ``self`` is indexed\n by partitions, and the degree of a basis element is the size of\n the partition indexing it.\n\n - ``cache_function`` -- a function which accepts an\n integer `n` as its input and creates the cache for that homogeneous\n component (saving it in ``cache_dict``).\n\n - ``cache_dict`` -- a dictionary storing a cache.\n It should be indexed by the positive integers `n`. Its values\n are dictionaries indexed by the partitions of size `n`. The values\n of those latter dictionaries are, again, dictionaries indexed by\n partitions of size `n`. Altogether, ``cache_dict`` should be\n understood to encode a graded linear map from `M` to the\n realization ``self`` of the ring of symmetric functions; the\n encoding is done in such a way that, for any `n` and any partitions\n ``lam`` and ``mu`` of `n`, the ``self[mu]``-coordinate of the image\n of ``M[lam]`` under this linear map (in the basis ``self``) is\n ``cache_dict[lam][mu]``.\n\n - ``subs_dict`` -- (optional) a dictionary for any substitutions\n to make after the value is extracted from ``cache_dict``.\n\n EXAMPLES::\n\n sage: R.<x> = QQ[]\n sage: Sym = SymmetricFunctions(R)\n sage: s = Sym.s()\n sage: p21 = Partition([2,1])\n sage: a = s(p21)\n sage: e = Sym.e()\n sage: cache_dict = {}\n sage: cache_dict[3] = {}\n sage: cache_dict[3][p21] = {}\n sage: cache_dict[3][p21][p21] = x^2\n sage: cache_dict[3][p21][Partition([1,1,1])] = 3*x\n sage: cache_function = lambda n: 0 #do nothing\n sage: e._from_cache(a, cache_function, cache_dict)\n 3*x*e[1, 1, 1] + x^2*e[2, 1]\n sage: e._from_cache(a, cache_function, cache_dict, x=2)\n 6*e[1, 1, 1] + 4*e[2, 1]\n '
BR = self.base_ring()
zero = BR.zero()
z_elt = {}
for (part, c) in element.monomial_coefficients().iteritems():
if (sum(part) not in cache_dict):
cache_function(sum(part))
part = _Partitions(part)
for (part2, c2) in cache_dict[sum(part)][part].iteritems():
if hasattr(c2, 'subs'):
c3 = (c * BR(c2.subs(**subs_dict)))
else:
c3 = (c * BR(c2))
z_elt[part2] = (z_elt.get(part2, zero) + BR(c3))
return self._from_dict(z_elt)<|docstring|>Return the image of an element ``element`` of some realization `M`
of the ring of symmetric functions under a linear map from `M` to
``self`` whose matrix representation is cached (with ``cache_dict``
serving as cache, and ``cache_function`` as the function which
precomputes this cache).
INPUT:
- ``element`` -- an element of a realization `M` of the ring of
symmetric functions. Note that `M` can be a different realization
than the one in which ``self`` is written, and does not have to
be specified. It is assumed that the basis of ``self`` is indexed
by partitions, and the degree of a basis element is the size of
the partition indexing it.
- ``cache_function`` -- a function which accepts an
integer `n` as its input and creates the cache for that homogeneous
component (saving it in ``cache_dict``).
- ``cache_dict`` -- a dictionary storing a cache.
It should be indexed by the positive integers `n`. Its values
are dictionaries indexed by the partitions of size `n`. The values
of those latter dictionaries are, again, dictionaries indexed by
partitions of size `n`. Altogether, ``cache_dict`` should be
understood to encode a graded linear map from `M` to the
realization ``self`` of the ring of symmetric functions; the
encoding is done in such a way that, for any `n` and any partitions
``lam`` and ``mu`` of `n`, the ``self[mu]``-coordinate of the image
of ``M[lam]`` under this linear map (in the basis ``self``) is
``cache_dict[lam][mu]``.
- ``subs_dict`` -- (optional) a dictionary for any substitutions
to make after the value is extracted from ``cache_dict``.
EXAMPLES::
sage: R.<x> = QQ[]
sage: Sym = SymmetricFunctions(R)
sage: s = Sym.s()
sage: p21 = Partition([2,1])
sage: a = s(p21)
sage: e = Sym.e()
sage: cache_dict = {}
sage: cache_dict[3] = {}
sage: cache_dict[3][p21] = {}
sage: cache_dict[3][p21][p21] = x^2
sage: cache_dict[3][p21][Partition([1,1,1])] = 3*x
sage: cache_function = lambda n: 0 #do nothing
sage: e._from_cache(a, cache_function, cache_dict)
3*x*e[1, 1, 1] + x^2*e[2, 1]
sage: e._from_cache(a, cache_function, cache_dict, x=2)
6*e[1, 1, 1] + 4*e[2, 1]<|endoftext|>
|
d703ff42f57b12651b0b16b352d161d63786e416398f5d1f04a2841f0749db4a
|
def _invert_morphism(self, n, base_ring, self_to_other_cache, other_to_self_cache, to_other_function=None, to_self_function=None, upper_triangular=False, lower_triangular=False, ones_on_diagonal=False):
"\n Compute the inverse of a morphism between ``self`` and ``other``\n (more precisely, its `n`-th graded component).\n\n In order to use this, you must be able to compute the morphism in\n one direction. This method assumes that the morphism is indeed\n invertible.\n\n INPUT:\n\n - ``n`` -- an integer, the homogeneous component of\n symmetric functions for which we want to a morphism's inverse\n\n - ``base_ring`` -- the base ring being worked over\n\n - ``self_to_other_cache`` -- a dictionary which\n stores the transition from ``self`` to ``other``\n\n - ``other_to_self_cache`` -- a dictionary which\n stores the transition from ``other`` to ``self``\n\n - ``to_other_function`` -- a function which takes in\n a partition and returns a function which gives the coefficients of\n ``self(part)`` in the ``other`` basis\n\n - ``to_self_function`` -- a function which takes in a\n partition and returns a function which gives the coefficients of\n ``other(part)`` in ``self``\n\n - ``upper_triangular`` -- a boolean, if ``True``, the\n inverse will be computed by back substitution\n\n - ``lower_triangular`` -- a boolean, if ``True``, the\n inverse will be computed by forward substitution\n\n - ``ones_on_diagonal`` -- a boolean, if ``True``, the\n entries on the diagonal of the morphism (and inverse) matrix are\n assumed to be ones. This is used to remove divisions from the\n forward and back substitute algorithms.\n\n OUTPUT:\n\n Nothing is returned, but the caches ``self_to_other_cache``\n and ``other_to_self_cache`` are updated with the `n`-th degree\n components of the respective transition matrices.\n\n EXAMPLES:\n\n First, we will do an example of inverting the morphism\n which sends a Schur function to its conjugate Schur function. Note\n that this is an involution. ::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: conj = lambda p1: lambda p2: QQ(1) if p2 == p1.conjugate() else QQ(0)\n sage: c1 = {}\n sage: c2 = {}\n sage: s._invert_morphism(4, QQ, c1, c2, to_other_function = conj)\n sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]\n sage: l(c1[4])\n [([1, 1, 1, 1], [([4], 1)]),\n ([2, 1, 1], [([3, 1], 1)]),\n ([2, 2], [([2, 2], 1)]),\n ([3, 1], [([2, 1, 1], 1)]),\n ([4], [([1, 1, 1, 1], 1)])]\n sage: l(c2[4])\n [([1, 1, 1, 1], [([4], 1)]),\n ([2, 1, 1], [([3, 1], 1)]),\n ([2, 2], [([2, 2], 1)]),\n ([3, 1], [([2, 1, 1], 1)]),\n ([4], [([1, 1, 1, 1], 1)])]\n sage: c2 == c1\n True\n\n We can check that we get the same results if we specify\n ``to_self_function = conj``::\n\n sage: d1 = {}\n sage: d2 = {}\n sage: s._invert_morphism(4, QQ, d1, d2, to_self_function = conj)\n sage: d1 == c1\n True\n sage: d2 == c2\n True\n\n Now we do an example of upper triangularity and check that we get\n the same thing whether or not we specify ``ones_on_diagonal``::\n\n sage: f = lambda p1: lambda p2: QQ(1) if p2 <= p1 else QQ(0)\n sage: c1 = {}\n sage: c2 = {}\n sage: s._invert_morphism(3, QQ, c1, c2, to_other_function = f, upper_triangular=True)\n sage: l(c1[3])\n [([1, 1, 1], [([1, 1, 1], 1)]),\n ([2, 1], [([1, 1, 1], 1), ([2, 1], 1)]),\n ([3], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)])]\n sage: l(c2[3])\n [([1, 1, 1], [([1, 1, 1], 1)]),\n ([2, 1], [([1, 1, 1], -1), ([2, 1], 1)]),\n ([3], [([2, 1], -1), ([3], 1)])]\n\n ::\n\n sage: d1 = {}\n sage: d2 = {}\n sage: s._invert_morphism(3, QQ, d1, d2, to_other_function = f,upper_triangular=True, ones_on_diagonal=True)\n sage: c1 == d1\n True\n sage: c2 == d2\n True\n\n Finally, we do the same thing for lower triangular matrices::\n\n sage: f = lambda p1: lambda p2: QQ(1) if p2 >= p1 else QQ(0)\n sage: c1 = {}\n sage: c2 = {}\n sage: s._invert_morphism(3, QQ, c1, c2, to_other_function = f, lower_triangular=True)\n sage: l(c1[3])\n [([1, 1, 1], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)]),\n ([2, 1], [([2, 1], 1), ([3], 1)]),\n ([3], [([3], 1)])]\n\n ::\n\n sage: l(c2[3])\n [([1, 1, 1], [([1, 1, 1], 1), ([2, 1], -1)]),\n ([2, 1], [([2, 1], 1), ([3], -1)]),\n ([3], [([3], 1)])]\n\n ::\n\n sage: d1 = {}\n sage: d2 = {}\n sage: s._invert_morphism(3, QQ, d1, d2, to_other_function = f,lower_triangular=True, ones_on_diagonal=True)\n sage: c1 == d1\n True\n sage: c2 == d2\n True\n "
if (to_other_function is not None):
known_cache = self_to_other_cache
unknown_cache = other_to_self_cache
known_function = to_other_function
else:
unknown_cache = self_to_other_cache
known_cache = other_to_self_cache
known_function = to_self_function
if ((n in known_cache) and (n in unknown_cache)):
return
one = base_ring.one()
zero = base_ring.zero()
pn = sage.combinat.partition.Partitions_n(n).list()
len_pn = len(pn)
known_cache_n = {}
known_matrix_n = matrix(base_ring, len_pn, len_pn)
unknown_cache_n = {}
for i in range(len_pn):
known_cache_part = {}
f = known_function(pn[i])
for j in range(len_pn):
if (lower_triangular and (j > i)):
break
if (upper_triangular and (i > j)):
continue
value = f(pn[j])
if (value != zero):
known_cache_part[pn[j]] = value
known_matrix_n[(i, j)] = value
known_cache_n[pn[i]] = known_cache_part
unknown_cache_n[pn[i]] = {}
if ((upper_triangular is not False) and (lower_triangular is not False)):
raise ValueError('only one of upper_triangular and lower_triangular can be specified')
elif (upper_triangular is not False):
inverse = copy(known_matrix_n.parent().zero_matrix())
delta = (lambda i: (lambda j: (one if (i == j) else zero)))
for column in range(len_pn):
e = delta(column)
x = ([0] * len_pn)
for i in range((len_pn - 1), (- 1), (- 1)):
value = e(i)
if (not ones_on_diagonal):
value /= known_matrix_n[(i, i)]
for j in range((i + 1), len_pn):
if ones_on_diagonal:
value -= (known_matrix_n[(i, j)] * x[j])
else:
value -= ((known_matrix_n[(i, j)] * x[j]) / known_matrix_n[(i, i)])
x[i] = value
for j in range((column + 1)):
if (x[j] != zero):
inverse[(j, column)] = x[j]
elif (lower_triangular is not False):
inverse = copy(known_matrix_n.parent().zero_matrix())
delta = (lambda i: (lambda j: (one if (i == j) else zero)))
for column in range(len_pn):
e = delta(column)
x = []
for i in range(len_pn):
value = e(i)
if (not ones_on_diagonal):
value /= known_matrix_n[(i, i)]
for j in range(len(x)):
if ones_on_diagonal:
value -= (known_matrix_n[(i, j)] * x[j])
else:
value -= ((known_matrix_n[(i, j)] * x[j]) / known_matrix_n[(i, i)])
x.append(value)
for j in range(column, len(x)):
if (x[j] != zero):
inverse[(j, column)] = x[j]
else:
inverse = (~ known_matrix_n)
for i in range(len_pn):
for j in range(len_pn):
if (inverse[(i, j)] != zero):
if hasattr(self, '_normalize_coefficients'):
unknown_cache_n[pn[i]][pn[j]] = self._normalize_coefficients(inverse[(i, j)])
else:
unknown_cache_n[pn[i]][pn[j]] = inverse[(i, j)]
known_cache[n] = known_cache_n
unknown_cache[n] = unknown_cache_n
|
Compute the inverse of a morphism between ``self`` and ``other``
(more precisely, its `n`-th graded component).
In order to use this, you must be able to compute the morphism in
one direction. This method assumes that the morphism is indeed
invertible.
INPUT:
- ``n`` -- an integer, the homogeneous component of
symmetric functions for which we want to a morphism's inverse
- ``base_ring`` -- the base ring being worked over
- ``self_to_other_cache`` -- a dictionary which
stores the transition from ``self`` to ``other``
- ``other_to_self_cache`` -- a dictionary which
stores the transition from ``other`` to ``self``
- ``to_other_function`` -- a function which takes in
a partition and returns a function which gives the coefficients of
``self(part)`` in the ``other`` basis
- ``to_self_function`` -- a function which takes in a
partition and returns a function which gives the coefficients of
``other(part)`` in ``self``
- ``upper_triangular`` -- a boolean, if ``True``, the
inverse will be computed by back substitution
- ``lower_triangular`` -- a boolean, if ``True``, the
inverse will be computed by forward substitution
- ``ones_on_diagonal`` -- a boolean, if ``True``, the
entries on the diagonal of the morphism (and inverse) matrix are
assumed to be ones. This is used to remove divisions from the
forward and back substitute algorithms.
OUTPUT:
Nothing is returned, but the caches ``self_to_other_cache``
and ``other_to_self_cache`` are updated with the `n`-th degree
components of the respective transition matrices.
EXAMPLES:
First, we will do an example of inverting the morphism
which sends a Schur function to its conjugate Schur function. Note
that this is an involution. ::
sage: s = SymmetricFunctions(QQ).s()
sage: conj = lambda p1: lambda p2: QQ(1) if p2 == p1.conjugate() else QQ(0)
sage: c1 = {}
sage: c2 = {}
sage: s._invert_morphism(4, QQ, c1, c2, to_other_function = conj)
sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]
sage: l(c1[4])
[([1, 1, 1, 1], [([4], 1)]),
([2, 1, 1], [([3, 1], 1)]),
([2, 2], [([2, 2], 1)]),
([3, 1], [([2, 1, 1], 1)]),
([4], [([1, 1, 1, 1], 1)])]
sage: l(c2[4])
[([1, 1, 1, 1], [([4], 1)]),
([2, 1, 1], [([3, 1], 1)]),
([2, 2], [([2, 2], 1)]),
([3, 1], [([2, 1, 1], 1)]),
([4], [([1, 1, 1, 1], 1)])]
sage: c2 == c1
True
We can check that we get the same results if we specify
``to_self_function = conj``::
sage: d1 = {}
sage: d2 = {}
sage: s._invert_morphism(4, QQ, d1, d2, to_self_function = conj)
sage: d1 == c1
True
sage: d2 == c2
True
Now we do an example of upper triangularity and check that we get
the same thing whether or not we specify ``ones_on_diagonal``::
sage: f = lambda p1: lambda p2: QQ(1) if p2 <= p1 else QQ(0)
sage: c1 = {}
sage: c2 = {}
sage: s._invert_morphism(3, QQ, c1, c2, to_other_function = f, upper_triangular=True)
sage: l(c1[3])
[([1, 1, 1], [([1, 1, 1], 1)]),
([2, 1], [([1, 1, 1], 1), ([2, 1], 1)]),
([3], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)])]
sage: l(c2[3])
[([1, 1, 1], [([1, 1, 1], 1)]),
([2, 1], [([1, 1, 1], -1), ([2, 1], 1)]),
([3], [([2, 1], -1), ([3], 1)])]
::
sage: d1 = {}
sage: d2 = {}
sage: s._invert_morphism(3, QQ, d1, d2, to_other_function = f,upper_triangular=True, ones_on_diagonal=True)
sage: c1 == d1
True
sage: c2 == d2
True
Finally, we do the same thing for lower triangular matrices::
sage: f = lambda p1: lambda p2: QQ(1) if p2 >= p1 else QQ(0)
sage: c1 = {}
sage: c2 = {}
sage: s._invert_morphism(3, QQ, c1, c2, to_other_function = f, lower_triangular=True)
sage: l(c1[3])
[([1, 1, 1], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)]),
([2, 1], [([2, 1], 1), ([3], 1)]),
([3], [([3], 1)])]
::
sage: l(c2[3])
[([1, 1, 1], [([1, 1, 1], 1), ([2, 1], -1)]),
([2, 1], [([2, 1], 1), ([3], -1)]),
([3], [([3], 1)])]
::
sage: d1 = {}
sage: d2 = {}
sage: s._invert_morphism(3, QQ, d1, d2, to_other_function = f,lower_triangular=True, ones_on_diagonal=True)
sage: c1 == d1
True
sage: c2 == d2
True
|
src/sage/combinat/sf/sfa.py
|
_invert_morphism
|
bopopescu/sagesmc
| 5 |
python
|
def _invert_morphism(self, n, base_ring, self_to_other_cache, other_to_self_cache, to_other_function=None, to_self_function=None, upper_triangular=False, lower_triangular=False, ones_on_diagonal=False):
"\n Compute the inverse of a morphism between ``self`` and ``other``\n (more precisely, its `n`-th graded component).\n\n In order to use this, you must be able to compute the morphism in\n one direction. This method assumes that the morphism is indeed\n invertible.\n\n INPUT:\n\n - ``n`` -- an integer, the homogeneous component of\n symmetric functions for which we want to a morphism's inverse\n\n - ``base_ring`` -- the base ring being worked over\n\n - ``self_to_other_cache`` -- a dictionary which\n stores the transition from ``self`` to ``other``\n\n - ``other_to_self_cache`` -- a dictionary which\n stores the transition from ``other`` to ``self``\n\n - ``to_other_function`` -- a function which takes in\n a partition and returns a function which gives the coefficients of\n ``self(part)`` in the ``other`` basis\n\n - ``to_self_function`` -- a function which takes in a\n partition and returns a function which gives the coefficients of\n ``other(part)`` in ``self``\n\n - ``upper_triangular`` -- a boolean, if ``True``, the\n inverse will be computed by back substitution\n\n - ``lower_triangular`` -- a boolean, if ``True``, the\n inverse will be computed by forward substitution\n\n - ``ones_on_diagonal`` -- a boolean, if ``True``, the\n entries on the diagonal of the morphism (and inverse) matrix are\n assumed to be ones. This is used to remove divisions from the\n forward and back substitute algorithms.\n\n OUTPUT:\n\n Nothing is returned, but the caches ``self_to_other_cache``\n and ``other_to_self_cache`` are updated with the `n`-th degree\n components of the respective transition matrices.\n\n EXAMPLES:\n\n First, we will do an example of inverting the morphism\n which sends a Schur function to its conjugate Schur function. Note\n that this is an involution. ::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: conj = lambda p1: lambda p2: QQ(1) if p2 == p1.conjugate() else QQ(0)\n sage: c1 = {}\n sage: c2 = {}\n sage: s._invert_morphism(4, QQ, c1, c2, to_other_function = conj)\n sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]\n sage: l(c1[4])\n [([1, 1, 1, 1], [([4], 1)]),\n ([2, 1, 1], [([3, 1], 1)]),\n ([2, 2], [([2, 2], 1)]),\n ([3, 1], [([2, 1, 1], 1)]),\n ([4], [([1, 1, 1, 1], 1)])]\n sage: l(c2[4])\n [([1, 1, 1, 1], [([4], 1)]),\n ([2, 1, 1], [([3, 1], 1)]),\n ([2, 2], [([2, 2], 1)]),\n ([3, 1], [([2, 1, 1], 1)]),\n ([4], [([1, 1, 1, 1], 1)])]\n sage: c2 == c1\n True\n\n We can check that we get the same results if we specify\n ``to_self_function = conj``::\n\n sage: d1 = {}\n sage: d2 = {}\n sage: s._invert_morphism(4, QQ, d1, d2, to_self_function = conj)\n sage: d1 == c1\n True\n sage: d2 == c2\n True\n\n Now we do an example of upper triangularity and check that we get\n the same thing whether or not we specify ``ones_on_diagonal``::\n\n sage: f = lambda p1: lambda p2: QQ(1) if p2 <= p1 else QQ(0)\n sage: c1 = {}\n sage: c2 = {}\n sage: s._invert_morphism(3, QQ, c1, c2, to_other_function = f, upper_triangular=True)\n sage: l(c1[3])\n [([1, 1, 1], [([1, 1, 1], 1)]),\n ([2, 1], [([1, 1, 1], 1), ([2, 1], 1)]),\n ([3], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)])]\n sage: l(c2[3])\n [([1, 1, 1], [([1, 1, 1], 1)]),\n ([2, 1], [([1, 1, 1], -1), ([2, 1], 1)]),\n ([3], [([2, 1], -1), ([3], 1)])]\n\n ::\n\n sage: d1 = {}\n sage: d2 = {}\n sage: s._invert_morphism(3, QQ, d1, d2, to_other_function = f,upper_triangular=True, ones_on_diagonal=True)\n sage: c1 == d1\n True\n sage: c2 == d2\n True\n\n Finally, we do the same thing for lower triangular matrices::\n\n sage: f = lambda p1: lambda p2: QQ(1) if p2 >= p1 else QQ(0)\n sage: c1 = {}\n sage: c2 = {}\n sage: s._invert_morphism(3, QQ, c1, c2, to_other_function = f, lower_triangular=True)\n sage: l(c1[3])\n [([1, 1, 1], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)]),\n ([2, 1], [([2, 1], 1), ([3], 1)]),\n ([3], [([3], 1)])]\n\n ::\n\n sage: l(c2[3])\n [([1, 1, 1], [([1, 1, 1], 1), ([2, 1], -1)]),\n ([2, 1], [([2, 1], 1), ([3], -1)]),\n ([3], [([3], 1)])]\n\n ::\n\n sage: d1 = {}\n sage: d2 = {}\n sage: s._invert_morphism(3, QQ, d1, d2, to_other_function = f,lower_triangular=True, ones_on_diagonal=True)\n sage: c1 == d1\n True\n sage: c2 == d2\n True\n "
if (to_other_function is not None):
known_cache = self_to_other_cache
unknown_cache = other_to_self_cache
known_function = to_other_function
else:
unknown_cache = self_to_other_cache
known_cache = other_to_self_cache
known_function = to_self_function
if ((n in known_cache) and (n in unknown_cache)):
return
one = base_ring.one()
zero = base_ring.zero()
pn = sage.combinat.partition.Partitions_n(n).list()
len_pn = len(pn)
known_cache_n = {}
known_matrix_n = matrix(base_ring, len_pn, len_pn)
unknown_cache_n = {}
for i in range(len_pn):
known_cache_part = {}
f = known_function(pn[i])
for j in range(len_pn):
if (lower_triangular and (j > i)):
break
if (upper_triangular and (i > j)):
continue
value = f(pn[j])
if (value != zero):
known_cache_part[pn[j]] = value
known_matrix_n[(i, j)] = value
known_cache_n[pn[i]] = known_cache_part
unknown_cache_n[pn[i]] = {}
if ((upper_triangular is not False) and (lower_triangular is not False)):
raise ValueError('only one of upper_triangular and lower_triangular can be specified')
elif (upper_triangular is not False):
inverse = copy(known_matrix_n.parent().zero_matrix())
delta = (lambda i: (lambda j: (one if (i == j) else zero)))
for column in range(len_pn):
e = delta(column)
x = ([0] * len_pn)
for i in range((len_pn - 1), (- 1), (- 1)):
value = e(i)
if (not ones_on_diagonal):
value /= known_matrix_n[(i, i)]
for j in range((i + 1), len_pn):
if ones_on_diagonal:
value -= (known_matrix_n[(i, j)] * x[j])
else:
value -= ((known_matrix_n[(i, j)] * x[j]) / known_matrix_n[(i, i)])
x[i] = value
for j in range((column + 1)):
if (x[j] != zero):
inverse[(j, column)] = x[j]
elif (lower_triangular is not False):
inverse = copy(known_matrix_n.parent().zero_matrix())
delta = (lambda i: (lambda j: (one if (i == j) else zero)))
for column in range(len_pn):
e = delta(column)
x = []
for i in range(len_pn):
value = e(i)
if (not ones_on_diagonal):
value /= known_matrix_n[(i, i)]
for j in range(len(x)):
if ones_on_diagonal:
value -= (known_matrix_n[(i, j)] * x[j])
else:
value -= ((known_matrix_n[(i, j)] * x[j]) / known_matrix_n[(i, i)])
x.append(value)
for j in range(column, len(x)):
if (x[j] != zero):
inverse[(j, column)] = x[j]
else:
inverse = (~ known_matrix_n)
for i in range(len_pn):
for j in range(len_pn):
if (inverse[(i, j)] != zero):
if hasattr(self, '_normalize_coefficients'):
unknown_cache_n[pn[i]][pn[j]] = self._normalize_coefficients(inverse[(i, j)])
else:
unknown_cache_n[pn[i]][pn[j]] = inverse[(i, j)]
known_cache[n] = known_cache_n
unknown_cache[n] = unknown_cache_n
|
def _invert_morphism(self, n, base_ring, self_to_other_cache, other_to_self_cache, to_other_function=None, to_self_function=None, upper_triangular=False, lower_triangular=False, ones_on_diagonal=False):
"\n Compute the inverse of a morphism between ``self`` and ``other``\n (more precisely, its `n`-th graded component).\n\n In order to use this, you must be able to compute the morphism in\n one direction. This method assumes that the morphism is indeed\n invertible.\n\n INPUT:\n\n - ``n`` -- an integer, the homogeneous component of\n symmetric functions for which we want to a morphism's inverse\n\n - ``base_ring`` -- the base ring being worked over\n\n - ``self_to_other_cache`` -- a dictionary which\n stores the transition from ``self`` to ``other``\n\n - ``other_to_self_cache`` -- a dictionary which\n stores the transition from ``other`` to ``self``\n\n - ``to_other_function`` -- a function which takes in\n a partition and returns a function which gives the coefficients of\n ``self(part)`` in the ``other`` basis\n\n - ``to_self_function`` -- a function which takes in a\n partition and returns a function which gives the coefficients of\n ``other(part)`` in ``self``\n\n - ``upper_triangular`` -- a boolean, if ``True``, the\n inverse will be computed by back substitution\n\n - ``lower_triangular`` -- a boolean, if ``True``, the\n inverse will be computed by forward substitution\n\n - ``ones_on_diagonal`` -- a boolean, if ``True``, the\n entries on the diagonal of the morphism (and inverse) matrix are\n assumed to be ones. This is used to remove divisions from the\n forward and back substitute algorithms.\n\n OUTPUT:\n\n Nothing is returned, but the caches ``self_to_other_cache``\n and ``other_to_self_cache`` are updated with the `n`-th degree\n components of the respective transition matrices.\n\n EXAMPLES:\n\n First, we will do an example of inverting the morphism\n which sends a Schur function to its conjugate Schur function. Note\n that this is an involution. ::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: conj = lambda p1: lambda p2: QQ(1) if p2 == p1.conjugate() else QQ(0)\n sage: c1 = {}\n sage: c2 = {}\n sage: s._invert_morphism(4, QQ, c1, c2, to_other_function = conj)\n sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]\n sage: l(c1[4])\n [([1, 1, 1, 1], [([4], 1)]),\n ([2, 1, 1], [([3, 1], 1)]),\n ([2, 2], [([2, 2], 1)]),\n ([3, 1], [([2, 1, 1], 1)]),\n ([4], [([1, 1, 1, 1], 1)])]\n sage: l(c2[4])\n [([1, 1, 1, 1], [([4], 1)]),\n ([2, 1, 1], [([3, 1], 1)]),\n ([2, 2], [([2, 2], 1)]),\n ([3, 1], [([2, 1, 1], 1)]),\n ([4], [([1, 1, 1, 1], 1)])]\n sage: c2 == c1\n True\n\n We can check that we get the same results if we specify\n ``to_self_function = conj``::\n\n sage: d1 = {}\n sage: d2 = {}\n sage: s._invert_morphism(4, QQ, d1, d2, to_self_function = conj)\n sage: d1 == c1\n True\n sage: d2 == c2\n True\n\n Now we do an example of upper triangularity and check that we get\n the same thing whether or not we specify ``ones_on_diagonal``::\n\n sage: f = lambda p1: lambda p2: QQ(1) if p2 <= p1 else QQ(0)\n sage: c1 = {}\n sage: c2 = {}\n sage: s._invert_morphism(3, QQ, c1, c2, to_other_function = f, upper_triangular=True)\n sage: l(c1[3])\n [([1, 1, 1], [([1, 1, 1], 1)]),\n ([2, 1], [([1, 1, 1], 1), ([2, 1], 1)]),\n ([3], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)])]\n sage: l(c2[3])\n [([1, 1, 1], [([1, 1, 1], 1)]),\n ([2, 1], [([1, 1, 1], -1), ([2, 1], 1)]),\n ([3], [([2, 1], -1), ([3], 1)])]\n\n ::\n\n sage: d1 = {}\n sage: d2 = {}\n sage: s._invert_morphism(3, QQ, d1, d2, to_other_function = f,upper_triangular=True, ones_on_diagonal=True)\n sage: c1 == d1\n True\n sage: c2 == d2\n True\n\n Finally, we do the same thing for lower triangular matrices::\n\n sage: f = lambda p1: lambda p2: QQ(1) if p2 >= p1 else QQ(0)\n sage: c1 = {}\n sage: c2 = {}\n sage: s._invert_morphism(3, QQ, c1, c2, to_other_function = f, lower_triangular=True)\n sage: l(c1[3])\n [([1, 1, 1], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)]),\n ([2, 1], [([2, 1], 1), ([3], 1)]),\n ([3], [([3], 1)])]\n\n ::\n\n sage: l(c2[3])\n [([1, 1, 1], [([1, 1, 1], 1), ([2, 1], -1)]),\n ([2, 1], [([2, 1], 1), ([3], -1)]),\n ([3], [([3], 1)])]\n\n ::\n\n sage: d1 = {}\n sage: d2 = {}\n sage: s._invert_morphism(3, QQ, d1, d2, to_other_function = f,lower_triangular=True, ones_on_diagonal=True)\n sage: c1 == d1\n True\n sage: c2 == d2\n True\n "
if (to_other_function is not None):
known_cache = self_to_other_cache
unknown_cache = other_to_self_cache
known_function = to_other_function
else:
unknown_cache = self_to_other_cache
known_cache = other_to_self_cache
known_function = to_self_function
if ((n in known_cache) and (n in unknown_cache)):
return
one = base_ring.one()
zero = base_ring.zero()
pn = sage.combinat.partition.Partitions_n(n).list()
len_pn = len(pn)
known_cache_n = {}
known_matrix_n = matrix(base_ring, len_pn, len_pn)
unknown_cache_n = {}
for i in range(len_pn):
known_cache_part = {}
f = known_function(pn[i])
for j in range(len_pn):
if (lower_triangular and (j > i)):
break
if (upper_triangular and (i > j)):
continue
value = f(pn[j])
if (value != zero):
known_cache_part[pn[j]] = value
known_matrix_n[(i, j)] = value
known_cache_n[pn[i]] = known_cache_part
unknown_cache_n[pn[i]] = {}
if ((upper_triangular is not False) and (lower_triangular is not False)):
raise ValueError('only one of upper_triangular and lower_triangular can be specified')
elif (upper_triangular is not False):
inverse = copy(known_matrix_n.parent().zero_matrix())
delta = (lambda i: (lambda j: (one if (i == j) else zero)))
for column in range(len_pn):
e = delta(column)
x = ([0] * len_pn)
for i in range((len_pn - 1), (- 1), (- 1)):
value = e(i)
if (not ones_on_diagonal):
value /= known_matrix_n[(i, i)]
for j in range((i + 1), len_pn):
if ones_on_diagonal:
value -= (known_matrix_n[(i, j)] * x[j])
else:
value -= ((known_matrix_n[(i, j)] * x[j]) / known_matrix_n[(i, i)])
x[i] = value
for j in range((column + 1)):
if (x[j] != zero):
inverse[(j, column)] = x[j]
elif (lower_triangular is not False):
inverse = copy(known_matrix_n.parent().zero_matrix())
delta = (lambda i: (lambda j: (one if (i == j) else zero)))
for column in range(len_pn):
e = delta(column)
x = []
for i in range(len_pn):
value = e(i)
if (not ones_on_diagonal):
value /= known_matrix_n[(i, i)]
for j in range(len(x)):
if ones_on_diagonal:
value -= (known_matrix_n[(i, j)] * x[j])
else:
value -= ((known_matrix_n[(i, j)] * x[j]) / known_matrix_n[(i, i)])
x.append(value)
for j in range(column, len(x)):
if (x[j] != zero):
inverse[(j, column)] = x[j]
else:
inverse = (~ known_matrix_n)
for i in range(len_pn):
for j in range(len_pn):
if (inverse[(i, j)] != zero):
if hasattr(self, '_normalize_coefficients'):
unknown_cache_n[pn[i]][pn[j]] = self._normalize_coefficients(inverse[(i, j)])
else:
unknown_cache_n[pn[i]][pn[j]] = inverse[(i, j)]
known_cache[n] = known_cache_n
unknown_cache[n] = unknown_cache_n<|docstring|>Compute the inverse of a morphism between ``self`` and ``other``
(more precisely, its `n`-th graded component).
In order to use this, you must be able to compute the morphism in
one direction. This method assumes that the morphism is indeed
invertible.
INPUT:
- ``n`` -- an integer, the homogeneous component of
symmetric functions for which we want to a morphism's inverse
- ``base_ring`` -- the base ring being worked over
- ``self_to_other_cache`` -- a dictionary which
stores the transition from ``self`` to ``other``
- ``other_to_self_cache`` -- a dictionary which
stores the transition from ``other`` to ``self``
- ``to_other_function`` -- a function which takes in
a partition and returns a function which gives the coefficients of
``self(part)`` in the ``other`` basis
- ``to_self_function`` -- a function which takes in a
partition and returns a function which gives the coefficients of
``other(part)`` in ``self``
- ``upper_triangular`` -- a boolean, if ``True``, the
inverse will be computed by back substitution
- ``lower_triangular`` -- a boolean, if ``True``, the
inverse will be computed by forward substitution
- ``ones_on_diagonal`` -- a boolean, if ``True``, the
entries on the diagonal of the morphism (and inverse) matrix are
assumed to be ones. This is used to remove divisions from the
forward and back substitute algorithms.
OUTPUT:
Nothing is returned, but the caches ``self_to_other_cache``
and ``other_to_self_cache`` are updated with the `n`-th degree
components of the respective transition matrices.
EXAMPLES:
First, we will do an example of inverting the morphism
which sends a Schur function to its conjugate Schur function. Note
that this is an involution. ::
sage: s = SymmetricFunctions(QQ).s()
sage: conj = lambda p1: lambda p2: QQ(1) if p2 == p1.conjugate() else QQ(0)
sage: c1 = {}
sage: c2 = {}
sage: s._invert_morphism(4, QQ, c1, c2, to_other_function = conj)
sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]
sage: l(c1[4])
[([1, 1, 1, 1], [([4], 1)]),
([2, 1, 1], [([3, 1], 1)]),
([2, 2], [([2, 2], 1)]),
([3, 1], [([2, 1, 1], 1)]),
([4], [([1, 1, 1, 1], 1)])]
sage: l(c2[4])
[([1, 1, 1, 1], [([4], 1)]),
([2, 1, 1], [([3, 1], 1)]),
([2, 2], [([2, 2], 1)]),
([3, 1], [([2, 1, 1], 1)]),
([4], [([1, 1, 1, 1], 1)])]
sage: c2 == c1
True
We can check that we get the same results if we specify
``to_self_function = conj``::
sage: d1 = {}
sage: d2 = {}
sage: s._invert_morphism(4, QQ, d1, d2, to_self_function = conj)
sage: d1 == c1
True
sage: d2 == c2
True
Now we do an example of upper triangularity and check that we get
the same thing whether or not we specify ``ones_on_diagonal``::
sage: f = lambda p1: lambda p2: QQ(1) if p2 <= p1 else QQ(0)
sage: c1 = {}
sage: c2 = {}
sage: s._invert_morphism(3, QQ, c1, c2, to_other_function = f, upper_triangular=True)
sage: l(c1[3])
[([1, 1, 1], [([1, 1, 1], 1)]),
([2, 1], [([1, 1, 1], 1), ([2, 1], 1)]),
([3], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)])]
sage: l(c2[3])
[([1, 1, 1], [([1, 1, 1], 1)]),
([2, 1], [([1, 1, 1], -1), ([2, 1], 1)]),
([3], [([2, 1], -1), ([3], 1)])]
::
sage: d1 = {}
sage: d2 = {}
sage: s._invert_morphism(3, QQ, d1, d2, to_other_function = f,upper_triangular=True, ones_on_diagonal=True)
sage: c1 == d1
True
sage: c2 == d2
True
Finally, we do the same thing for lower triangular matrices::
sage: f = lambda p1: lambda p2: QQ(1) if p2 >= p1 else QQ(0)
sage: c1 = {}
sage: c2 = {}
sage: s._invert_morphism(3, QQ, c1, c2, to_other_function = f, lower_triangular=True)
sage: l(c1[3])
[([1, 1, 1], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)]),
([2, 1], [([2, 1], 1), ([3], 1)]),
([3], [([3], 1)])]
::
sage: l(c2[3])
[([1, 1, 1], [([1, 1, 1], 1), ([2, 1], -1)]),
([2, 1], [([2, 1], 1), ([3], -1)]),
([3], [([3], 1)])]
::
sage: d1 = {}
sage: d2 = {}
sage: s._invert_morphism(3, QQ, d1, d2, to_other_function = f,lower_triangular=True, ones_on_diagonal=True)
sage: c1 == d1
True
sage: c2 == d2
True<|endoftext|>
|
0bd71aa1cc0e559c9b843be7dec8d6cfba938043d2ad2477c4721f44eab20f8c
|
def symmetric_function_ring(self):
"\n Return the family of symmetric functions associated to the\n basis ``self``.\n\n OUTPUT:\n\n - returns an instance of the ring of symmetric functions\n\n EXAMPLES::\n\n sage: schur = SymmetricFunctions(QQ).schur()\n sage: schur.symmetric_function_ring()\n Symmetric Functions over Rational Field\n sage: power = SymmetricFunctions(QQ['t']).power()\n sage: power.symmetric_function_ring()\n Symmetric Functions over Univariate Polynomial Ring in t over Rational Field\n "
return self.realization_of()
|
Return the family of symmetric functions associated to the
basis ``self``.
OUTPUT:
- returns an instance of the ring of symmetric functions
EXAMPLES::
sage: schur = SymmetricFunctions(QQ).schur()
sage: schur.symmetric_function_ring()
Symmetric Functions over Rational Field
sage: power = SymmetricFunctions(QQ['t']).power()
sage: power.symmetric_function_ring()
Symmetric Functions over Univariate Polynomial Ring in t over Rational Field
|
src/sage/combinat/sf/sfa.py
|
symmetric_function_ring
|
bopopescu/sagesmc
| 5 |
python
|
def symmetric_function_ring(self):
"\n Return the family of symmetric functions associated to the\n basis ``self``.\n\n OUTPUT:\n\n - returns an instance of the ring of symmetric functions\n\n EXAMPLES::\n\n sage: schur = SymmetricFunctions(QQ).schur()\n sage: schur.symmetric_function_ring()\n Symmetric Functions over Rational Field\n sage: power = SymmetricFunctions(QQ['t']).power()\n sage: power.symmetric_function_ring()\n Symmetric Functions over Univariate Polynomial Ring in t over Rational Field\n "
return self.realization_of()
|
def symmetric_function_ring(self):
"\n Return the family of symmetric functions associated to the\n basis ``self``.\n\n OUTPUT:\n\n - returns an instance of the ring of symmetric functions\n\n EXAMPLES::\n\n sage: schur = SymmetricFunctions(QQ).schur()\n sage: schur.symmetric_function_ring()\n Symmetric Functions over Rational Field\n sage: power = SymmetricFunctions(QQ['t']).power()\n sage: power.symmetric_function_ring()\n Symmetric Functions over Univariate Polynomial Ring in t over Rational Field\n "
return self.realization_of()<|docstring|>Return the family of symmetric functions associated to the
basis ``self``.
OUTPUT:
- returns an instance of the ring of symmetric functions
EXAMPLES::
sage: schur = SymmetricFunctions(QQ).schur()
sage: schur.symmetric_function_ring()
Symmetric Functions over Rational Field
sage: power = SymmetricFunctions(QQ['t']).power()
sage: power.symmetric_function_ring()
Symmetric Functions over Univariate Polynomial Ring in t over Rational Field<|endoftext|>
|
69f360a5dfada7070324b12353a9cb6671b44714dfd612f83e2f54822fc5a271
|
def prefix(self):
"\n Return the prefix on the elements of ``self``.\n\n EXAMPLES::\n\n sage: schur = SymmetricFunctions(QQ).schur()\n sage: schur([3,2,1])\n s[3, 2, 1]\n sage: schur.prefix()\n 's'\n "
return self._prefix
|
Return the prefix on the elements of ``self``.
EXAMPLES::
sage: schur = SymmetricFunctions(QQ).schur()
sage: schur([3,2,1])
s[3, 2, 1]
sage: schur.prefix()
's'
|
src/sage/combinat/sf/sfa.py
|
prefix
|
bopopescu/sagesmc
| 5 |
python
|
def prefix(self):
"\n Return the prefix on the elements of ``self``.\n\n EXAMPLES::\n\n sage: schur = SymmetricFunctions(QQ).schur()\n sage: schur([3,2,1])\n s[3, 2, 1]\n sage: schur.prefix()\n 's'\n "
return self._prefix
|
def prefix(self):
"\n Return the prefix on the elements of ``self``.\n\n EXAMPLES::\n\n sage: schur = SymmetricFunctions(QQ).schur()\n sage: schur([3,2,1])\n s[3, 2, 1]\n sage: schur.prefix()\n 's'\n "
return self._prefix<|docstring|>Return the prefix on the elements of ``self``.
EXAMPLES::
sage: schur = SymmetricFunctions(QQ).schur()
sage: schur([3,2,1])
s[3, 2, 1]
sage: schur.prefix()
's'<|endoftext|>
|
342c62019f81fc2cf07b19199d462d7384e0378705df4e5bdbefb167b9afd688
|
def transition_matrix(self, basis, n):
'\n Return the transition matrix between ``self`` and ``basis`` for the\n homogeneous component of degree ``n``.\n\n INPUT:\n\n - ``basis`` -- a basis of the ring of symmetric functions\n - ``n`` -- a nonnegative integer\n\n OUTPUT:\n\n - a matrix of coefficients giving the expansion of the\n homogeneous degree-`n` elements of ``self`` in the\n degree-`n` elements of ``basis``\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: m = SymmetricFunctions(QQ).m()\n sage: s.transition_matrix(m,5)\n [1 1 1 1 1 1 1]\n [0 1 1 2 2 3 4]\n [0 0 1 1 2 3 5]\n [0 0 0 1 1 3 6]\n [0 0 0 0 1 2 5]\n [0 0 0 0 0 1 4]\n [0 0 0 0 0 0 1]\n sage: s.transition_matrix(m,1)\n [1]\n sage: s.transition_matrix(m,0)\n [1]\n\n ::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: s.transition_matrix(p, 4)\n [ 1/4 1/3 1/8 1/4 1/24]\n [-1/4 0 -1/8 1/4 1/8]\n [ 0 -1/3 1/4 0 1/12]\n [ 1/4 0 -1/8 -1/4 1/8]\n [-1/4 1/3 1/8 -1/4 1/24]\n sage: StoP = s.transition_matrix(p,4)\n sage: a = s([3,1])+5*s([1,1,1,1])-s([4])\n sage: a\n 5*s[1, 1, 1, 1] + s[3, 1] - s[4]\n sage: mon = a.support()\n sage: coeffs = a.coefficients()\n sage: coeffs\n [5, 1, -1]\n sage: mon\n [[1, 1, 1, 1], [3, 1], [4]]\n sage: cm = matrix([[-1,1,0,0,5]])\n sage: cm * StoP\n [-7/4 4/3 3/8 -5/4 7/24]\n sage: p(a)\n 7/24*p[1, 1, 1, 1] - 5/4*p[2, 1, 1] + 3/8*p[2, 2] + 4/3*p[3, 1] - 7/4*p[4]\n\n ::\n\n sage: h = SymmetricFunctions(QQ).h()\n sage: e = SymmetricFunctions(QQ).e()\n sage: s.transition_matrix(m,7) == h.transition_matrix(s,7).transpose()\n True\n\n ::\n\n sage: h.transition_matrix(m, 7) == h.transition_matrix(m, 7).transpose()\n True\n\n ::\n\n sage: h.transition_matrix(e, 7) == e.transition_matrix(h, 7)\n True\n\n ::\n\n sage: p.transition_matrix(s, 5)\n [ 1 -1 0 1 0 -1 1]\n [ 1 0 -1 0 1 0 -1]\n [ 1 -1 1 0 -1 1 -1]\n [ 1 1 -1 0 -1 1 1]\n [ 1 0 1 -2 1 0 1]\n [ 1 2 1 0 -1 -2 -1]\n [ 1 4 5 6 5 4 1]\n\n ::\n\n sage: e.transition_matrix(m,7) == e.transition_matrix(m,7).transpose()\n True\n '
P = sage.combinat.partition.Partitions_n(n)
Plist = P.list()
m = []
for row_part in Plist:
z = basis(self(row_part))
m.append(map((lambda col_part: z.coefficient(col_part)), Plist))
return matrix(m)
|
Return the transition matrix between ``self`` and ``basis`` for the
homogeneous component of degree ``n``.
INPUT:
- ``basis`` -- a basis of the ring of symmetric functions
- ``n`` -- a nonnegative integer
OUTPUT:
- a matrix of coefficients giving the expansion of the
homogeneous degree-`n` elements of ``self`` in the
degree-`n` elements of ``basis``
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: m = SymmetricFunctions(QQ).m()
sage: s.transition_matrix(m,5)
[1 1 1 1 1 1 1]
[0 1 1 2 2 3 4]
[0 0 1 1 2 3 5]
[0 0 0 1 1 3 6]
[0 0 0 0 1 2 5]
[0 0 0 0 0 1 4]
[0 0 0 0 0 0 1]
sage: s.transition_matrix(m,1)
[1]
sage: s.transition_matrix(m,0)
[1]
::
sage: p = SymmetricFunctions(QQ).p()
sage: s.transition_matrix(p, 4)
[ 1/4 1/3 1/8 1/4 1/24]
[-1/4 0 -1/8 1/4 1/8]
[ 0 -1/3 1/4 0 1/12]
[ 1/4 0 -1/8 -1/4 1/8]
[-1/4 1/3 1/8 -1/4 1/24]
sage: StoP = s.transition_matrix(p,4)
sage: a = s([3,1])+5*s([1,1,1,1])-s([4])
sage: a
5*s[1, 1, 1, 1] + s[3, 1] - s[4]
sage: mon = a.support()
sage: coeffs = a.coefficients()
sage: coeffs
[5, 1, -1]
sage: mon
[[1, 1, 1, 1], [3, 1], [4]]
sage: cm = matrix([[-1,1,0,0,5]])
sage: cm * StoP
[-7/4 4/3 3/8 -5/4 7/24]
sage: p(a)
7/24*p[1, 1, 1, 1] - 5/4*p[2, 1, 1] + 3/8*p[2, 2] + 4/3*p[3, 1] - 7/4*p[4]
::
sage: h = SymmetricFunctions(QQ).h()
sage: e = SymmetricFunctions(QQ).e()
sage: s.transition_matrix(m,7) == h.transition_matrix(s,7).transpose()
True
::
sage: h.transition_matrix(m, 7) == h.transition_matrix(m, 7).transpose()
True
::
sage: h.transition_matrix(e, 7) == e.transition_matrix(h, 7)
True
::
sage: p.transition_matrix(s, 5)
[ 1 -1 0 1 0 -1 1]
[ 1 0 -1 0 1 0 -1]
[ 1 -1 1 0 -1 1 -1]
[ 1 1 -1 0 -1 1 1]
[ 1 0 1 -2 1 0 1]
[ 1 2 1 0 -1 -2 -1]
[ 1 4 5 6 5 4 1]
::
sage: e.transition_matrix(m,7) == e.transition_matrix(m,7).transpose()
True
|
src/sage/combinat/sf/sfa.py
|
transition_matrix
|
bopopescu/sagesmc
| 5 |
python
|
def transition_matrix(self, basis, n):
'\n Return the transition matrix between ``self`` and ``basis`` for the\n homogeneous component of degree ``n``.\n\n INPUT:\n\n - ``basis`` -- a basis of the ring of symmetric functions\n - ``n`` -- a nonnegative integer\n\n OUTPUT:\n\n - a matrix of coefficients giving the expansion of the\n homogeneous degree-`n` elements of ``self`` in the\n degree-`n` elements of ``basis``\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: m = SymmetricFunctions(QQ).m()\n sage: s.transition_matrix(m,5)\n [1 1 1 1 1 1 1]\n [0 1 1 2 2 3 4]\n [0 0 1 1 2 3 5]\n [0 0 0 1 1 3 6]\n [0 0 0 0 1 2 5]\n [0 0 0 0 0 1 4]\n [0 0 0 0 0 0 1]\n sage: s.transition_matrix(m,1)\n [1]\n sage: s.transition_matrix(m,0)\n [1]\n\n ::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: s.transition_matrix(p, 4)\n [ 1/4 1/3 1/8 1/4 1/24]\n [-1/4 0 -1/8 1/4 1/8]\n [ 0 -1/3 1/4 0 1/12]\n [ 1/4 0 -1/8 -1/4 1/8]\n [-1/4 1/3 1/8 -1/4 1/24]\n sage: StoP = s.transition_matrix(p,4)\n sage: a = s([3,1])+5*s([1,1,1,1])-s([4])\n sage: a\n 5*s[1, 1, 1, 1] + s[3, 1] - s[4]\n sage: mon = a.support()\n sage: coeffs = a.coefficients()\n sage: coeffs\n [5, 1, -1]\n sage: mon\n [[1, 1, 1, 1], [3, 1], [4]]\n sage: cm = matrix([[-1,1,0,0,5]])\n sage: cm * StoP\n [-7/4 4/3 3/8 -5/4 7/24]\n sage: p(a)\n 7/24*p[1, 1, 1, 1] - 5/4*p[2, 1, 1] + 3/8*p[2, 2] + 4/3*p[3, 1] - 7/4*p[4]\n\n ::\n\n sage: h = SymmetricFunctions(QQ).h()\n sage: e = SymmetricFunctions(QQ).e()\n sage: s.transition_matrix(m,7) == h.transition_matrix(s,7).transpose()\n True\n\n ::\n\n sage: h.transition_matrix(m, 7) == h.transition_matrix(m, 7).transpose()\n True\n\n ::\n\n sage: h.transition_matrix(e, 7) == e.transition_matrix(h, 7)\n True\n\n ::\n\n sage: p.transition_matrix(s, 5)\n [ 1 -1 0 1 0 -1 1]\n [ 1 0 -1 0 1 0 -1]\n [ 1 -1 1 0 -1 1 -1]\n [ 1 1 -1 0 -1 1 1]\n [ 1 0 1 -2 1 0 1]\n [ 1 2 1 0 -1 -2 -1]\n [ 1 4 5 6 5 4 1]\n\n ::\n\n sage: e.transition_matrix(m,7) == e.transition_matrix(m,7).transpose()\n True\n '
P = sage.combinat.partition.Partitions_n(n)
Plist = P.list()
m = []
for row_part in Plist:
z = basis(self(row_part))
m.append(map((lambda col_part: z.coefficient(col_part)), Plist))
return matrix(m)
|
def transition_matrix(self, basis, n):
'\n Return the transition matrix between ``self`` and ``basis`` for the\n homogeneous component of degree ``n``.\n\n INPUT:\n\n - ``basis`` -- a basis of the ring of symmetric functions\n - ``n`` -- a nonnegative integer\n\n OUTPUT:\n\n - a matrix of coefficients giving the expansion of the\n homogeneous degree-`n` elements of ``self`` in the\n degree-`n` elements of ``basis``\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: m = SymmetricFunctions(QQ).m()\n sage: s.transition_matrix(m,5)\n [1 1 1 1 1 1 1]\n [0 1 1 2 2 3 4]\n [0 0 1 1 2 3 5]\n [0 0 0 1 1 3 6]\n [0 0 0 0 1 2 5]\n [0 0 0 0 0 1 4]\n [0 0 0 0 0 0 1]\n sage: s.transition_matrix(m,1)\n [1]\n sage: s.transition_matrix(m,0)\n [1]\n\n ::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: s.transition_matrix(p, 4)\n [ 1/4 1/3 1/8 1/4 1/24]\n [-1/4 0 -1/8 1/4 1/8]\n [ 0 -1/3 1/4 0 1/12]\n [ 1/4 0 -1/8 -1/4 1/8]\n [-1/4 1/3 1/8 -1/4 1/24]\n sage: StoP = s.transition_matrix(p,4)\n sage: a = s([3,1])+5*s([1,1,1,1])-s([4])\n sage: a\n 5*s[1, 1, 1, 1] + s[3, 1] - s[4]\n sage: mon = a.support()\n sage: coeffs = a.coefficients()\n sage: coeffs\n [5, 1, -1]\n sage: mon\n [[1, 1, 1, 1], [3, 1], [4]]\n sage: cm = matrix([[-1,1,0,0,5]])\n sage: cm * StoP\n [-7/4 4/3 3/8 -5/4 7/24]\n sage: p(a)\n 7/24*p[1, 1, 1, 1] - 5/4*p[2, 1, 1] + 3/8*p[2, 2] + 4/3*p[3, 1] - 7/4*p[4]\n\n ::\n\n sage: h = SymmetricFunctions(QQ).h()\n sage: e = SymmetricFunctions(QQ).e()\n sage: s.transition_matrix(m,7) == h.transition_matrix(s,7).transpose()\n True\n\n ::\n\n sage: h.transition_matrix(m, 7) == h.transition_matrix(m, 7).transpose()\n True\n\n ::\n\n sage: h.transition_matrix(e, 7) == e.transition_matrix(h, 7)\n True\n\n ::\n\n sage: p.transition_matrix(s, 5)\n [ 1 -1 0 1 0 -1 1]\n [ 1 0 -1 0 1 0 -1]\n [ 1 -1 1 0 -1 1 -1]\n [ 1 1 -1 0 -1 1 1]\n [ 1 0 1 -2 1 0 1]\n [ 1 2 1 0 -1 -2 -1]\n [ 1 4 5 6 5 4 1]\n\n ::\n\n sage: e.transition_matrix(m,7) == e.transition_matrix(m,7).transpose()\n True\n '
P = sage.combinat.partition.Partitions_n(n)
Plist = P.list()
m = []
for row_part in Plist:
z = basis(self(row_part))
m.append(map((lambda col_part: z.coefficient(col_part)), Plist))
return matrix(m)<|docstring|>Return the transition matrix between ``self`` and ``basis`` for the
homogeneous component of degree ``n``.
INPUT:
- ``basis`` -- a basis of the ring of symmetric functions
- ``n`` -- a nonnegative integer
OUTPUT:
- a matrix of coefficients giving the expansion of the
homogeneous degree-`n` elements of ``self`` in the
degree-`n` elements of ``basis``
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: m = SymmetricFunctions(QQ).m()
sage: s.transition_matrix(m,5)
[1 1 1 1 1 1 1]
[0 1 1 2 2 3 4]
[0 0 1 1 2 3 5]
[0 0 0 1 1 3 6]
[0 0 0 0 1 2 5]
[0 0 0 0 0 1 4]
[0 0 0 0 0 0 1]
sage: s.transition_matrix(m,1)
[1]
sage: s.transition_matrix(m,0)
[1]
::
sage: p = SymmetricFunctions(QQ).p()
sage: s.transition_matrix(p, 4)
[ 1/4 1/3 1/8 1/4 1/24]
[-1/4 0 -1/8 1/4 1/8]
[ 0 -1/3 1/4 0 1/12]
[ 1/4 0 -1/8 -1/4 1/8]
[-1/4 1/3 1/8 -1/4 1/24]
sage: StoP = s.transition_matrix(p,4)
sage: a = s([3,1])+5*s([1,1,1,1])-s([4])
sage: a
5*s[1, 1, 1, 1] + s[3, 1] - s[4]
sage: mon = a.support()
sage: coeffs = a.coefficients()
sage: coeffs
[5, 1, -1]
sage: mon
[[1, 1, 1, 1], [3, 1], [4]]
sage: cm = matrix([[-1,1,0,0,5]])
sage: cm * StoP
[-7/4 4/3 3/8 -5/4 7/24]
sage: p(a)
7/24*p[1, 1, 1, 1] - 5/4*p[2, 1, 1] + 3/8*p[2, 2] + 4/3*p[3, 1] - 7/4*p[4]
::
sage: h = SymmetricFunctions(QQ).h()
sage: e = SymmetricFunctions(QQ).e()
sage: s.transition_matrix(m,7) == h.transition_matrix(s,7).transpose()
True
::
sage: h.transition_matrix(m, 7) == h.transition_matrix(m, 7).transpose()
True
::
sage: h.transition_matrix(e, 7) == e.transition_matrix(h, 7)
True
::
sage: p.transition_matrix(s, 5)
[ 1 -1 0 1 0 -1 1]
[ 1 0 -1 0 1 0 -1]
[ 1 -1 1 0 -1 1 -1]
[ 1 1 -1 0 -1 1 1]
[ 1 0 1 -2 1 0 1]
[ 1 2 1 0 -1 -2 -1]
[ 1 4 5 6 5 4 1]
::
sage: e.transition_matrix(m,7) == e.transition_matrix(m,7).transpose()
True<|endoftext|>
|
3827c72ae4dd4d8bc42b48f58be2f910d272ae665ff03cbcdab5b08c9fdae004
|
def _gram_schmidt(self, n, source, scalar, cache, leading_coeff=None, upper_triangular=True):
"\n Apply Gram-Schmidt to ``source`` with respect to the scalar product\n ``scalar`` for all partitions of `n`. The scalar product is supposed\n to make the power-sum basis orthogonal. The Gram-Schmidt algorithm\n computes an orthogonal basis (with respect to the scalar product\n given by ``scalar``) of the `n`-th homogeneous component of the\n ring of symmetric functions such that the transition matrix from\n the basis ``source`` to this orthogonal basis is triangular.\n\n The result is not returned, but instead, the coefficients of the\n elements of the orthogonal basis with respect to the basis\n ``source`` are stored in the cache ``cache``.\n\n The implementation uses the powersum basis, so this function\n shouldn't be used unless the base ring is a `\\QQ`-algebra\n (or ``self`` and ``source`` are both the powersum basis).\n\n INPUT:\n\n - ``n`` -- nonnegative integer which specifies the size of\n the partitions\n - ``source`` -- a basis of the ring of symmetric functions\n - ``scalar`` -- a function ``zee`` from partitions to the base ring\n which specifies the scalar product by `\\langle p_{\\lambda},\n p_{\\lambda} \\rangle = \\mathrm{zee}(\\lambda)`.\n - ``cache`` -- a cache function\n - ``leading_coeff`` -- (default: ``None``) specifies the leading\n coefficients for Gram-Schmidt\n - ``upper_triangular`` -- (defaults to ``True``) boolean, indicates\n whether the transition is upper triangular or not\n\n EXAMPLES::\n\n sage: cache = {}\n sage: from sage.combinat.sf.sfa import zee\n sage: s = SymmetricFunctions(QQ).s()\n sage: m = SymmetricFunctions(QQ).m()\n sage: s._gram_schmidt(3, m, zee, cache)\n sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]\n sage: l(cache)\n [([1, 1, 1], [([1, 1, 1], 1)]),\n ([2, 1], [([1, 1, 1], 2), ([2, 1], 1)]),\n ([3], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)])]\n "
BR = self.base_ring()
one = BR.one()
p = self.realization_of().p()
pscalar = (lambda x, y: p._apply_multi_module_morphism(p(x), p(y), (lambda a, b: scalar(a)), orthogonal=True))
if (leading_coeff is None):
leading_coeff = (lambda x: one)
l = Partitions(n).list()
if upper_triangular:
l.reverse()
precomputed_elements = []
cache[l[0]] = {l[0]: leading_coeff(l[0])}
precomputed_elements.append((leading_coeff(l[0]) * source(l[0])))
for i in range(1, len(l)):
start = (leading_coeff(l[i]) * source(l[i]))
sub = 0
for j in range(i):
sub += ((pscalar(start, precomputed_elements[j]) / pscalar(precomputed_elements[j], precomputed_elements[j])) * precomputed_elements[j])
res = (start - sub)
if hasattr(self, '_normalize_coefficients'):
res = res.map_coefficients(self._normalize_coefficients)
precomputed_elements.append(res)
cache[l[i]] = {}
for j in range((i + 1)):
cache[l[i]][l[j]] = res.coefficient(l[j])
|
Apply Gram-Schmidt to ``source`` with respect to the scalar product
``scalar`` for all partitions of `n`. The scalar product is supposed
to make the power-sum basis orthogonal. The Gram-Schmidt algorithm
computes an orthogonal basis (with respect to the scalar product
given by ``scalar``) of the `n`-th homogeneous component of the
ring of symmetric functions such that the transition matrix from
the basis ``source`` to this orthogonal basis is triangular.
The result is not returned, but instead, the coefficients of the
elements of the orthogonal basis with respect to the basis
``source`` are stored in the cache ``cache``.
The implementation uses the powersum basis, so this function
shouldn't be used unless the base ring is a `\QQ`-algebra
(or ``self`` and ``source`` are both the powersum basis).
INPUT:
- ``n`` -- nonnegative integer which specifies the size of
the partitions
- ``source`` -- a basis of the ring of symmetric functions
- ``scalar`` -- a function ``zee`` from partitions to the base ring
which specifies the scalar product by `\langle p_{\lambda},
p_{\lambda} \rangle = \mathrm{zee}(\lambda)`.
- ``cache`` -- a cache function
- ``leading_coeff`` -- (default: ``None``) specifies the leading
coefficients for Gram-Schmidt
- ``upper_triangular`` -- (defaults to ``True``) boolean, indicates
whether the transition is upper triangular or not
EXAMPLES::
sage: cache = {}
sage: from sage.combinat.sf.sfa import zee
sage: s = SymmetricFunctions(QQ).s()
sage: m = SymmetricFunctions(QQ).m()
sage: s._gram_schmidt(3, m, zee, cache)
sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]
sage: l(cache)
[([1, 1, 1], [([1, 1, 1], 1)]),
([2, 1], [([1, 1, 1], 2), ([2, 1], 1)]),
([3], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)])]
|
src/sage/combinat/sf/sfa.py
|
_gram_schmidt
|
bopopescu/sagesmc
| 5 |
python
|
def _gram_schmidt(self, n, source, scalar, cache, leading_coeff=None, upper_triangular=True):
"\n Apply Gram-Schmidt to ``source`` with respect to the scalar product\n ``scalar`` for all partitions of `n`. The scalar product is supposed\n to make the power-sum basis orthogonal. The Gram-Schmidt algorithm\n computes an orthogonal basis (with respect to the scalar product\n given by ``scalar``) of the `n`-th homogeneous component of the\n ring of symmetric functions such that the transition matrix from\n the basis ``source`` to this orthogonal basis is triangular.\n\n The result is not returned, but instead, the coefficients of the\n elements of the orthogonal basis with respect to the basis\n ``source`` are stored in the cache ``cache``.\n\n The implementation uses the powersum basis, so this function\n shouldn't be used unless the base ring is a `\\QQ`-algebra\n (or ``self`` and ``source`` are both the powersum basis).\n\n INPUT:\n\n - ``n`` -- nonnegative integer which specifies the size of\n the partitions\n - ``source`` -- a basis of the ring of symmetric functions\n - ``scalar`` -- a function ``zee`` from partitions to the base ring\n which specifies the scalar product by `\\langle p_{\\lambda},\n p_{\\lambda} \\rangle = \\mathrm{zee}(\\lambda)`.\n - ``cache`` -- a cache function\n - ``leading_coeff`` -- (default: ``None``) specifies the leading\n coefficients for Gram-Schmidt\n - ``upper_triangular`` -- (defaults to ``True``) boolean, indicates\n whether the transition is upper triangular or not\n\n EXAMPLES::\n\n sage: cache = {}\n sage: from sage.combinat.sf.sfa import zee\n sage: s = SymmetricFunctions(QQ).s()\n sage: m = SymmetricFunctions(QQ).m()\n sage: s._gram_schmidt(3, m, zee, cache)\n sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]\n sage: l(cache)\n [([1, 1, 1], [([1, 1, 1], 1)]),\n ([2, 1], [([1, 1, 1], 2), ([2, 1], 1)]),\n ([3], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)])]\n "
BR = self.base_ring()
one = BR.one()
p = self.realization_of().p()
pscalar = (lambda x, y: p._apply_multi_module_morphism(p(x), p(y), (lambda a, b: scalar(a)), orthogonal=True))
if (leading_coeff is None):
leading_coeff = (lambda x: one)
l = Partitions(n).list()
if upper_triangular:
l.reverse()
precomputed_elements = []
cache[l[0]] = {l[0]: leading_coeff(l[0])}
precomputed_elements.append((leading_coeff(l[0]) * source(l[0])))
for i in range(1, len(l)):
start = (leading_coeff(l[i]) * source(l[i]))
sub = 0
for j in range(i):
sub += ((pscalar(start, precomputed_elements[j]) / pscalar(precomputed_elements[j], precomputed_elements[j])) * precomputed_elements[j])
res = (start - sub)
if hasattr(self, '_normalize_coefficients'):
res = res.map_coefficients(self._normalize_coefficients)
precomputed_elements.append(res)
cache[l[i]] = {}
for j in range((i + 1)):
cache[l[i]][l[j]] = res.coefficient(l[j])
|
def _gram_schmidt(self, n, source, scalar, cache, leading_coeff=None, upper_triangular=True):
"\n Apply Gram-Schmidt to ``source`` with respect to the scalar product\n ``scalar`` for all partitions of `n`. The scalar product is supposed\n to make the power-sum basis orthogonal. The Gram-Schmidt algorithm\n computes an orthogonal basis (with respect to the scalar product\n given by ``scalar``) of the `n`-th homogeneous component of the\n ring of symmetric functions such that the transition matrix from\n the basis ``source`` to this orthogonal basis is triangular.\n\n The result is not returned, but instead, the coefficients of the\n elements of the orthogonal basis with respect to the basis\n ``source`` are stored in the cache ``cache``.\n\n The implementation uses the powersum basis, so this function\n shouldn't be used unless the base ring is a `\\QQ`-algebra\n (or ``self`` and ``source`` are both the powersum basis).\n\n INPUT:\n\n - ``n`` -- nonnegative integer which specifies the size of\n the partitions\n - ``source`` -- a basis of the ring of symmetric functions\n - ``scalar`` -- a function ``zee`` from partitions to the base ring\n which specifies the scalar product by `\\langle p_{\\lambda},\n p_{\\lambda} \\rangle = \\mathrm{zee}(\\lambda)`.\n - ``cache`` -- a cache function\n - ``leading_coeff`` -- (default: ``None``) specifies the leading\n coefficients for Gram-Schmidt\n - ``upper_triangular`` -- (defaults to ``True``) boolean, indicates\n whether the transition is upper triangular or not\n\n EXAMPLES::\n\n sage: cache = {}\n sage: from sage.combinat.sf.sfa import zee\n sage: s = SymmetricFunctions(QQ).s()\n sage: m = SymmetricFunctions(QQ).m()\n sage: s._gram_schmidt(3, m, zee, cache)\n sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]\n sage: l(cache)\n [([1, 1, 1], [([1, 1, 1], 1)]),\n ([2, 1], [([1, 1, 1], 2), ([2, 1], 1)]),\n ([3], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)])]\n "
BR = self.base_ring()
one = BR.one()
p = self.realization_of().p()
pscalar = (lambda x, y: p._apply_multi_module_morphism(p(x), p(y), (lambda a, b: scalar(a)), orthogonal=True))
if (leading_coeff is None):
leading_coeff = (lambda x: one)
l = Partitions(n).list()
if upper_triangular:
l.reverse()
precomputed_elements = []
cache[l[0]] = {l[0]: leading_coeff(l[0])}
precomputed_elements.append((leading_coeff(l[0]) * source(l[0])))
for i in range(1, len(l)):
start = (leading_coeff(l[i]) * source(l[i]))
sub = 0
for j in range(i):
sub += ((pscalar(start, precomputed_elements[j]) / pscalar(precomputed_elements[j], precomputed_elements[j])) * precomputed_elements[j])
res = (start - sub)
if hasattr(self, '_normalize_coefficients'):
res = res.map_coefficients(self._normalize_coefficients)
precomputed_elements.append(res)
cache[l[i]] = {}
for j in range((i + 1)):
cache[l[i]][l[j]] = res.coefficient(l[j])<|docstring|>Apply Gram-Schmidt to ``source`` with respect to the scalar product
``scalar`` for all partitions of `n`. The scalar product is supposed
to make the power-sum basis orthogonal. The Gram-Schmidt algorithm
computes an orthogonal basis (with respect to the scalar product
given by ``scalar``) of the `n`-th homogeneous component of the
ring of symmetric functions such that the transition matrix from
the basis ``source`` to this orthogonal basis is triangular.
The result is not returned, but instead, the coefficients of the
elements of the orthogonal basis with respect to the basis
``source`` are stored in the cache ``cache``.
The implementation uses the powersum basis, so this function
shouldn't be used unless the base ring is a `\QQ`-algebra
(or ``self`` and ``source`` are both the powersum basis).
INPUT:
- ``n`` -- nonnegative integer which specifies the size of
the partitions
- ``source`` -- a basis of the ring of symmetric functions
- ``scalar`` -- a function ``zee`` from partitions to the base ring
which specifies the scalar product by `\langle p_{\lambda},
p_{\lambda} \rangle = \mathrm{zee}(\lambda)`.
- ``cache`` -- a cache function
- ``leading_coeff`` -- (default: ``None``) specifies the leading
coefficients for Gram-Schmidt
- ``upper_triangular`` -- (defaults to ``True``) boolean, indicates
whether the transition is upper triangular or not
EXAMPLES::
sage: cache = {}
sage: from sage.combinat.sf.sfa import zee
sage: s = SymmetricFunctions(QQ).s()
sage: m = SymmetricFunctions(QQ).m()
sage: s._gram_schmidt(3, m, zee, cache)
sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]
sage: l(cache)
[([1, 1, 1], [([1, 1, 1], 1)]),
([2, 1], [([1, 1, 1], 2), ([2, 1], 1)]),
([3], [([1, 1, 1], 1), ([2, 1], 1), ([3], 1)])]<|endoftext|>
|
dd2e917d5cc382d1f2493bf70393e0116d661a3dab8facdd3afbc471ac45eead
|
def _inner_plethysm_pk_g(self, k, g, cache):
'\n Return the inner plethysm between the power-sum symmetric\n function `p_k` and the symmetric function ``g``.\n\n See :meth:`inner_plethysm` for the definition of inner\n plethysm.\n\n .. WARNING::\n\n The function ``g`` *must* be given in the power-sum\n basis for this method to return a correct result.\n\n ALGORITHM:\n\n Express ``g`` in the power sum basis as\n `g = \\sum_\\mu c_\\mu p_\\mu/z_\\mu`\n (where `z_\\mu` is the size of the centralizer of any\n permutation with cycle type `\\mu`). Then, the inner plethysm\n is calculated as\n\n .. MATH::\n\n p_k \\{ g \\} = \\sum_\\mu c_\\mu p_k \\{ p_\\mu/z_\\mu \\}~.\n\n The inner plethysm `p_k \\{ p_mu/z_\\mu \\}` is given by the formula\n\n .. MATH::\n\n p_k \\{ p_\\mu/z_\\mu \\} = \\sum_{\\nu : \\nu^k = \\mu } p_{\\nu}/z_{\\nu}~,\n\n where `\\nu^k` is the `k`-th power of `nu` (see\n :~sage.combinat.partition.partition_power`).\n\n .. SEEALSO:: :func:`~sage.combinat.partition.partition_power`,\n :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.inner_plethysm`\n\n INPUT:\n\n - ``k`` -- a positive integer\n\n - ``g`` -- a symmetric function in the power sum basis\n\n - ``cache`` -- a dictionary whose keys are (k, g) pairs\n and values are the cached output of this function\n\n EXAMPLES::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: p._inner_plethysm_pk_g(2, p([1,1,1]), {})\n p[1, 1, 1] + 3*p[2, 1]\n sage: p._inner_plethysm_pk_g(5, p([2,2,1,1,1]), {})\n p[2, 2, 1, 1, 1]\n '
try:
return cache[(k, g)]
except KeyError:
pass
p = self.realization_of().p()
res = 0
degrees = uniq([sum(m) for m in g.support()])
for d in degrees:
for mu in sage.combinat.partition.Partitions(d):
mu_k = mu.power(k)
if (mu_k in g):
res += (((g.coefficient(mu_k) * mu_k.centralizer_size()) / mu.centralizer_size()) * p(mu))
cache[(k, g)] = res
return res
|
Return the inner plethysm between the power-sum symmetric
function `p_k` and the symmetric function ``g``.
See :meth:`inner_plethysm` for the definition of inner
plethysm.
.. WARNING::
The function ``g`` *must* be given in the power-sum
basis for this method to return a correct result.
ALGORITHM:
Express ``g`` in the power sum basis as
`g = \sum_\mu c_\mu p_\mu/z_\mu`
(where `z_\mu` is the size of the centralizer of any
permutation with cycle type `\mu`). Then, the inner plethysm
is calculated as
.. MATH::
p_k \{ g \} = \sum_\mu c_\mu p_k \{ p_\mu/z_\mu \}~.
The inner plethysm `p_k \{ p_mu/z_\mu \}` is given by the formula
.. MATH::
p_k \{ p_\mu/z_\mu \} = \sum_{\nu : \nu^k = \mu } p_{\nu}/z_{\nu}~,
where `\nu^k` is the `k`-th power of `nu` (see
:~sage.combinat.partition.partition_power`).
.. SEEALSO:: :func:`~sage.combinat.partition.partition_power`,
:meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.inner_plethysm`
INPUT:
- ``k`` -- a positive integer
- ``g`` -- a symmetric function in the power sum basis
- ``cache`` -- a dictionary whose keys are (k, g) pairs
and values are the cached output of this function
EXAMPLES::
sage: p = SymmetricFunctions(QQ).p()
sage: p._inner_plethysm_pk_g(2, p([1,1,1]), {})
p[1, 1, 1] + 3*p[2, 1]
sage: p._inner_plethysm_pk_g(5, p([2,2,1,1,1]), {})
p[2, 2, 1, 1, 1]
|
src/sage/combinat/sf/sfa.py
|
_inner_plethysm_pk_g
|
bopopescu/sagesmc
| 5 |
python
|
def _inner_plethysm_pk_g(self, k, g, cache):
'\n Return the inner plethysm between the power-sum symmetric\n function `p_k` and the symmetric function ``g``.\n\n See :meth:`inner_plethysm` for the definition of inner\n plethysm.\n\n .. WARNING::\n\n The function ``g`` *must* be given in the power-sum\n basis for this method to return a correct result.\n\n ALGORITHM:\n\n Express ``g`` in the power sum basis as\n `g = \\sum_\\mu c_\\mu p_\\mu/z_\\mu`\n (where `z_\\mu` is the size of the centralizer of any\n permutation with cycle type `\\mu`). Then, the inner plethysm\n is calculated as\n\n .. MATH::\n\n p_k \\{ g \\} = \\sum_\\mu c_\\mu p_k \\{ p_\\mu/z_\\mu \\}~.\n\n The inner plethysm `p_k \\{ p_mu/z_\\mu \\}` is given by the formula\n\n .. MATH::\n\n p_k \\{ p_\\mu/z_\\mu \\} = \\sum_{\\nu : \\nu^k = \\mu } p_{\\nu}/z_{\\nu}~,\n\n where `\\nu^k` is the `k`-th power of `nu` (see\n :~sage.combinat.partition.partition_power`).\n\n .. SEEALSO:: :func:`~sage.combinat.partition.partition_power`,\n :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.inner_plethysm`\n\n INPUT:\n\n - ``k`` -- a positive integer\n\n - ``g`` -- a symmetric function in the power sum basis\n\n - ``cache`` -- a dictionary whose keys are (k, g) pairs\n and values are the cached output of this function\n\n EXAMPLES::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: p._inner_plethysm_pk_g(2, p([1,1,1]), {})\n p[1, 1, 1] + 3*p[2, 1]\n sage: p._inner_plethysm_pk_g(5, p([2,2,1,1,1]), {})\n p[2, 2, 1, 1, 1]\n '
try:
return cache[(k, g)]
except KeyError:
pass
p = self.realization_of().p()
res = 0
degrees = uniq([sum(m) for m in g.support()])
for d in degrees:
for mu in sage.combinat.partition.Partitions(d):
mu_k = mu.power(k)
if (mu_k in g):
res += (((g.coefficient(mu_k) * mu_k.centralizer_size()) / mu.centralizer_size()) * p(mu))
cache[(k, g)] = res
return res
|
def _inner_plethysm_pk_g(self, k, g, cache):
'\n Return the inner plethysm between the power-sum symmetric\n function `p_k` and the symmetric function ``g``.\n\n See :meth:`inner_plethysm` for the definition of inner\n plethysm.\n\n .. WARNING::\n\n The function ``g`` *must* be given in the power-sum\n basis for this method to return a correct result.\n\n ALGORITHM:\n\n Express ``g`` in the power sum basis as\n `g = \\sum_\\mu c_\\mu p_\\mu/z_\\mu`\n (where `z_\\mu` is the size of the centralizer of any\n permutation with cycle type `\\mu`). Then, the inner plethysm\n is calculated as\n\n .. MATH::\n\n p_k \\{ g \\} = \\sum_\\mu c_\\mu p_k \\{ p_\\mu/z_\\mu \\}~.\n\n The inner plethysm `p_k \\{ p_mu/z_\\mu \\}` is given by the formula\n\n .. MATH::\n\n p_k \\{ p_\\mu/z_\\mu \\} = \\sum_{\\nu : \\nu^k = \\mu } p_{\\nu}/z_{\\nu}~,\n\n where `\\nu^k` is the `k`-th power of `nu` (see\n :~sage.combinat.partition.partition_power`).\n\n .. SEEALSO:: :func:`~sage.combinat.partition.partition_power`,\n :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.inner_plethysm`\n\n INPUT:\n\n - ``k`` -- a positive integer\n\n - ``g`` -- a symmetric function in the power sum basis\n\n - ``cache`` -- a dictionary whose keys are (k, g) pairs\n and values are the cached output of this function\n\n EXAMPLES::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: p._inner_plethysm_pk_g(2, p([1,1,1]), {})\n p[1, 1, 1] + 3*p[2, 1]\n sage: p._inner_plethysm_pk_g(5, p([2,2,1,1,1]), {})\n p[2, 2, 1, 1, 1]\n '
try:
return cache[(k, g)]
except KeyError:
pass
p = self.realization_of().p()
res = 0
degrees = uniq([sum(m) for m in g.support()])
for d in degrees:
for mu in sage.combinat.partition.Partitions(d):
mu_k = mu.power(k)
if (mu_k in g):
res += (((g.coefficient(mu_k) * mu_k.centralizer_size()) / mu.centralizer_size()) * p(mu))
cache[(k, g)] = res
return res<|docstring|>Return the inner plethysm between the power-sum symmetric
function `p_k` and the symmetric function ``g``.
See :meth:`inner_plethysm` for the definition of inner
plethysm.
.. WARNING::
The function ``g`` *must* be given in the power-sum
basis for this method to return a correct result.
ALGORITHM:
Express ``g`` in the power sum basis as
`g = \sum_\mu c_\mu p_\mu/z_\mu`
(where `z_\mu` is the size of the centralizer of any
permutation with cycle type `\mu`). Then, the inner plethysm
is calculated as
.. MATH::
p_k \{ g \} = \sum_\mu c_\mu p_k \{ p_\mu/z_\mu \}~.
The inner plethysm `p_k \{ p_mu/z_\mu \}` is given by the formula
.. MATH::
p_k \{ p_\mu/z_\mu \} = \sum_{\nu : \nu^k = \mu } p_{\nu}/z_{\nu}~,
where `\nu^k` is the `k`-th power of `nu` (see
:~sage.combinat.partition.partition_power`).
.. SEEALSO:: :func:`~sage.combinat.partition.partition_power`,
:meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.inner_plethysm`
INPUT:
- ``k`` -- a positive integer
- ``g`` -- a symmetric function in the power sum basis
- ``cache`` -- a dictionary whose keys are (k, g) pairs
and values are the cached output of this function
EXAMPLES::
sage: p = SymmetricFunctions(QQ).p()
sage: p._inner_plethysm_pk_g(2, p([1,1,1]), {})
p[1, 1, 1] + 3*p[2, 1]
sage: p._inner_plethysm_pk_g(5, p([2,2,1,1,1]), {})
p[2, 2, 1, 1, 1]<|endoftext|>
|
34df7963a90079e476cc42c88648b1f634aa3ac1feaa10ac64a87a68f9da235a
|
def _inner_plethysm_pnu_g(self, p_x, cache, nu):
'\n Return the inner plethysm of the power-sum symmetric function\n `p_\\nu` with another symmetric function ``p_x`` in the\n power-sum basis.\n\n See :meth:`inner_plethysm` for the definition of inner\n plethysm.\n\n .. WARNING::\n\n The function ``p_x`` *must* be given in the power-sum\n basis for this method to return a correct result.\n\n The computation uses the inner plethysm of `p_k` and ``p_x``\n and the identity\n\n .. MATH::\n\n (f \\cdot g) \\{ h \\} = (f \\{ h \\}) \\ast (g \\{ h \\})~.\n\n .. SEEALSO:: :meth:`_inner_plethysm_pk_g`, \n :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.itensor`,\n :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.inner_plethysm`\n\n INPUT:\n\n - ``p_x`` -- a symmetric function in the power sum basis\n\n - ``cache`` -- a cache function\n\n - ``nu`` -- a partition\n\n Note that the order of the arguments is somewhat strange in order\n to facilitate partial function application.\n\n OUTPUT:\n\n - an element of the basis ``self``\n\n EXAMPLES::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: s = SymmetricFunctions(QQ).s()\n sage: p._inner_plethysm_pnu_g( p([1,1,1]), {}, Partition([2,1]))\n 6*p[1, 1, 1]\n sage: p._inner_plethysm_pnu_g( p([1,1,1]), {}, Partition([]))\n 1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3]\n sage: s(_)\n s[3]\n '
if (len(nu) == 0):
s = self.realization_of().s()
degrees = [part.size() for part in p_x.support()]
degrees = uniq(degrees)
if (0 in degrees):
ext = self([])
else:
ext = 0
return (ext + self(sum([s([n]) for n in degrees if (n != 0)])))
res = [self._inner_plethysm_pk_g(k, p_x, cache) for k in nu]
return self(reduce((lambda x, y: (0 if (x == 0) else x.itensor(y))), res))
|
Return the inner plethysm of the power-sum symmetric function
`p_\nu` with another symmetric function ``p_x`` in the
power-sum basis.
See :meth:`inner_plethysm` for the definition of inner
plethysm.
.. WARNING::
The function ``p_x`` *must* be given in the power-sum
basis for this method to return a correct result.
The computation uses the inner plethysm of `p_k` and ``p_x``
and the identity
.. MATH::
(f \cdot g) \{ h \} = (f \{ h \}) \ast (g \{ h \})~.
.. SEEALSO:: :meth:`_inner_plethysm_pk_g`,
:meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.itensor`,
:meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.inner_plethysm`
INPUT:
- ``p_x`` -- a symmetric function in the power sum basis
- ``cache`` -- a cache function
- ``nu`` -- a partition
Note that the order of the arguments is somewhat strange in order
to facilitate partial function application.
OUTPUT:
- an element of the basis ``self``
EXAMPLES::
sage: p = SymmetricFunctions(QQ).p()
sage: s = SymmetricFunctions(QQ).s()
sage: p._inner_plethysm_pnu_g( p([1,1,1]), {}, Partition([2,1]))
6*p[1, 1, 1]
sage: p._inner_plethysm_pnu_g( p([1,1,1]), {}, Partition([]))
1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3]
sage: s(_)
s[3]
|
src/sage/combinat/sf/sfa.py
|
_inner_plethysm_pnu_g
|
bopopescu/sagesmc
| 5 |
python
|
def _inner_plethysm_pnu_g(self, p_x, cache, nu):
'\n Return the inner plethysm of the power-sum symmetric function\n `p_\\nu` with another symmetric function ``p_x`` in the\n power-sum basis.\n\n See :meth:`inner_plethysm` for the definition of inner\n plethysm.\n\n .. WARNING::\n\n The function ``p_x`` *must* be given in the power-sum\n basis for this method to return a correct result.\n\n The computation uses the inner plethysm of `p_k` and ``p_x``\n and the identity\n\n .. MATH::\n\n (f \\cdot g) \\{ h \\} = (f \\{ h \\}) \\ast (g \\{ h \\})~.\n\n .. SEEALSO:: :meth:`_inner_plethysm_pk_g`, \n :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.itensor`,\n :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.inner_plethysm`\n\n INPUT:\n\n - ``p_x`` -- a symmetric function in the power sum basis\n\n - ``cache`` -- a cache function\n\n - ``nu`` -- a partition\n\n Note that the order of the arguments is somewhat strange in order\n to facilitate partial function application.\n\n OUTPUT:\n\n - an element of the basis ``self``\n\n EXAMPLES::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: s = SymmetricFunctions(QQ).s()\n sage: p._inner_plethysm_pnu_g( p([1,1,1]), {}, Partition([2,1]))\n 6*p[1, 1, 1]\n sage: p._inner_plethysm_pnu_g( p([1,1,1]), {}, Partition([]))\n 1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3]\n sage: s(_)\n s[3]\n '
if (len(nu) == 0):
s = self.realization_of().s()
degrees = [part.size() for part in p_x.support()]
degrees = uniq(degrees)
if (0 in degrees):
ext = self([])
else:
ext = 0
return (ext + self(sum([s([n]) for n in degrees if (n != 0)])))
res = [self._inner_plethysm_pk_g(k, p_x, cache) for k in nu]
return self(reduce((lambda x, y: (0 if (x == 0) else x.itensor(y))), res))
|
def _inner_plethysm_pnu_g(self, p_x, cache, nu):
'\n Return the inner plethysm of the power-sum symmetric function\n `p_\\nu` with another symmetric function ``p_x`` in the\n power-sum basis.\n\n See :meth:`inner_plethysm` for the definition of inner\n plethysm.\n\n .. WARNING::\n\n The function ``p_x`` *must* be given in the power-sum\n basis for this method to return a correct result.\n\n The computation uses the inner plethysm of `p_k` and ``p_x``\n and the identity\n\n .. MATH::\n\n (f \\cdot g) \\{ h \\} = (f \\{ h \\}) \\ast (g \\{ h \\})~.\n\n .. SEEALSO:: :meth:`_inner_plethysm_pk_g`, \n :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.itensor`,\n :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.inner_plethysm`\n\n INPUT:\n\n - ``p_x`` -- a symmetric function in the power sum basis\n\n - ``cache`` -- a cache function\n\n - ``nu`` -- a partition\n\n Note that the order of the arguments is somewhat strange in order\n to facilitate partial function application.\n\n OUTPUT:\n\n - an element of the basis ``self``\n\n EXAMPLES::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: s = SymmetricFunctions(QQ).s()\n sage: p._inner_plethysm_pnu_g( p([1,1,1]), {}, Partition([2,1]))\n 6*p[1, 1, 1]\n sage: p._inner_plethysm_pnu_g( p([1,1,1]), {}, Partition([]))\n 1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3]\n sage: s(_)\n s[3]\n '
if (len(nu) == 0):
s = self.realization_of().s()
degrees = [part.size() for part in p_x.support()]
degrees = uniq(degrees)
if (0 in degrees):
ext = self([])
else:
ext = 0
return (ext + self(sum([s([n]) for n in degrees if (n != 0)])))
res = [self._inner_plethysm_pk_g(k, p_x, cache) for k in nu]
return self(reduce((lambda x, y: (0 if (x == 0) else x.itensor(y))), res))<|docstring|>Return the inner plethysm of the power-sum symmetric function
`p_\nu` with another symmetric function ``p_x`` in the
power-sum basis.
See :meth:`inner_plethysm` for the definition of inner
plethysm.
.. WARNING::
The function ``p_x`` *must* be given in the power-sum
basis for this method to return a correct result.
The computation uses the inner plethysm of `p_k` and ``p_x``
and the identity
.. MATH::
(f \cdot g) \{ h \} = (f \{ h \}) \ast (g \{ h \})~.
.. SEEALSO:: :meth:`_inner_plethysm_pk_g`,
:meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.itensor`,
:meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.inner_plethysm`
INPUT:
- ``p_x`` -- a symmetric function in the power sum basis
- ``cache`` -- a cache function
- ``nu`` -- a partition
Note that the order of the arguments is somewhat strange in order
to facilitate partial function application.
OUTPUT:
- an element of the basis ``self``
EXAMPLES::
sage: p = SymmetricFunctions(QQ).p()
sage: s = SymmetricFunctions(QQ).s()
sage: p._inner_plethysm_pnu_g( p([1,1,1]), {}, Partition([2,1]))
6*p[1, 1, 1]
sage: p._inner_plethysm_pnu_g( p([1,1,1]), {}, Partition([]))
1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3]
sage: s(_)
s[3]<|endoftext|>
|
6c2f099336f25f8accdbd4806200eac996b16a7b147ed879f7220aa3d03e66ad
|
def _dual_basis_default(self):
'\n Returns the default value for ``self.dual_basis()``\n\n .. SEEALSO:: :meth:`dual_basis`\n\n EXAMPLES:\n\n This default implementation constructs the dual basis using\n the standard (Hall) scalar product::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: Sym.p()._dual_basis_default()\n Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product\n\n This is meant to be overriden by subclasses for which an\n explicit dual basis is known::\n\n sage: Sym.s()._dual_basis_default()\n Symmetric Functions over Rational Field in the Schur basis\n sage: Sym.h()._dual_basis_default()\n Symmetric Functions over Rational Field in the monomial basis\n sage: Sym.m()._dual_basis_default()\n Symmetric Functions over Rational Field in the homogeneous basis\n sage: Sym.f()._dual_basis_default()\n Symmetric Functions over Rational Field in the elementary basis\n sage: Sym.e()._dual_basis_default()\n Symmetric Functions over Rational Field in the forgotten basis\n sage: Sym.f()._dual_basis_default()\n Symmetric Functions over Rational Field in the elementary basis\n '
return self.dual_basis(scalar=zee, scalar_name='Hall scalar product')
|
Returns the default value for ``self.dual_basis()``
.. SEEALSO:: :meth:`dual_basis`
EXAMPLES:
This default implementation constructs the dual basis using
the standard (Hall) scalar product::
sage: Sym = SymmetricFunctions(QQ)
sage: Sym.p()._dual_basis_default()
Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product
This is meant to be overriden by subclasses for which an
explicit dual basis is known::
sage: Sym.s()._dual_basis_default()
Symmetric Functions over Rational Field in the Schur basis
sage: Sym.h()._dual_basis_default()
Symmetric Functions over Rational Field in the monomial basis
sage: Sym.m()._dual_basis_default()
Symmetric Functions over Rational Field in the homogeneous basis
sage: Sym.f()._dual_basis_default()
Symmetric Functions over Rational Field in the elementary basis
sage: Sym.e()._dual_basis_default()
Symmetric Functions over Rational Field in the forgotten basis
sage: Sym.f()._dual_basis_default()
Symmetric Functions over Rational Field in the elementary basis
|
src/sage/combinat/sf/sfa.py
|
_dual_basis_default
|
bopopescu/sagesmc
| 5 |
python
|
def _dual_basis_default(self):
'\n Returns the default value for ``self.dual_basis()``\n\n .. SEEALSO:: :meth:`dual_basis`\n\n EXAMPLES:\n\n This default implementation constructs the dual basis using\n the standard (Hall) scalar product::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: Sym.p()._dual_basis_default()\n Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product\n\n This is meant to be overriden by subclasses for which an\n explicit dual basis is known::\n\n sage: Sym.s()._dual_basis_default()\n Symmetric Functions over Rational Field in the Schur basis\n sage: Sym.h()._dual_basis_default()\n Symmetric Functions over Rational Field in the monomial basis\n sage: Sym.m()._dual_basis_default()\n Symmetric Functions over Rational Field in the homogeneous basis\n sage: Sym.f()._dual_basis_default()\n Symmetric Functions over Rational Field in the elementary basis\n sage: Sym.e()._dual_basis_default()\n Symmetric Functions over Rational Field in the forgotten basis\n sage: Sym.f()._dual_basis_default()\n Symmetric Functions over Rational Field in the elementary basis\n '
return self.dual_basis(scalar=zee, scalar_name='Hall scalar product')
|
def _dual_basis_default(self):
'\n Returns the default value for ``self.dual_basis()``\n\n .. SEEALSO:: :meth:`dual_basis`\n\n EXAMPLES:\n\n This default implementation constructs the dual basis using\n the standard (Hall) scalar product::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: Sym.p()._dual_basis_default()\n Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product\n\n This is meant to be overriden by subclasses for which an\n explicit dual basis is known::\n\n sage: Sym.s()._dual_basis_default()\n Symmetric Functions over Rational Field in the Schur basis\n sage: Sym.h()._dual_basis_default()\n Symmetric Functions over Rational Field in the monomial basis\n sage: Sym.m()._dual_basis_default()\n Symmetric Functions over Rational Field in the homogeneous basis\n sage: Sym.f()._dual_basis_default()\n Symmetric Functions over Rational Field in the elementary basis\n sage: Sym.e()._dual_basis_default()\n Symmetric Functions over Rational Field in the forgotten basis\n sage: Sym.f()._dual_basis_default()\n Symmetric Functions over Rational Field in the elementary basis\n '
return self.dual_basis(scalar=zee, scalar_name='Hall scalar product')<|docstring|>Returns the default value for ``self.dual_basis()``
.. SEEALSO:: :meth:`dual_basis`
EXAMPLES:
This default implementation constructs the dual basis using
the standard (Hall) scalar product::
sage: Sym = SymmetricFunctions(QQ)
sage: Sym.p()._dual_basis_default()
Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product
This is meant to be overriden by subclasses for which an
explicit dual basis is known::
sage: Sym.s()._dual_basis_default()
Symmetric Functions over Rational Field in the Schur basis
sage: Sym.h()._dual_basis_default()
Symmetric Functions over Rational Field in the monomial basis
sage: Sym.m()._dual_basis_default()
Symmetric Functions over Rational Field in the homogeneous basis
sage: Sym.f()._dual_basis_default()
Symmetric Functions over Rational Field in the elementary basis
sage: Sym.e()._dual_basis_default()
Symmetric Functions over Rational Field in the forgotten basis
sage: Sym.f()._dual_basis_default()
Symmetric Functions over Rational Field in the elementary basis<|endoftext|>
|
37e2ca44bdfcd31a1ba5589651dae4320b535b1968d068eb5e3bdb9d3de56ba3
|
def dual_basis(self, scalar=None, scalar_name='', basis_name=None, prefix=None):
"\n Return the dual basis of ``self`` with respect to the scalar\n product ``scalar``.\n\n INPUT:\n\n - ``scalar`` -- A function ``zee`` from partitions to the base ring\n which specifies the scalar product by `\\langle p_{\\lambda},\n p_{\\lambda} \\rangle = \\mathrm{zee}(\\lambda)`. (Independently on the\n function chosen, the power sum basis will always be orthogonal;\n the function ``scalar`` only determines the norms of the basis\n elements.) If ``scalar`` is None, then the standard (Hall) scalar\n product is used.\n - ``scalar_name`` -- name of the scalar function\n - ``prefix`` -- prefix used to display the basis\n\n EXAMPLES:\n\n The duals of the elementary symmetric functions with respect to the\n Hall scalar product are the forgotten symmetric functions.\n\n ::\n\n sage: e = SymmetricFunctions(QQ).e()\n sage: f = e.dual_basis(prefix='f'); f\n Dual basis to Symmetric Functions over Rational Field in the elementary basis with respect to the Hall scalar product\n sage: f([2,1])^2\n 4*f[2, 2, 1, 1] + 6*f[2, 2, 2] + 2*f[3, 2, 1] + 2*f[3, 3] + 2*f[4, 1, 1] + f[4, 2]\n sage: f([2,1]).scalar(e([2,1]))\n 1\n sage: f([2,1]).scalar(e([1,1,1]))\n 0\n\n Since the power-sum symmetric functions are orthogonal, their duals\n with respect to the Hall scalar product are scalar multiples of\n themselves.\n\n ::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: q = p.dual_basis(prefix='q'); q\n Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product\n sage: q([2,1])^2\n 4*q[2, 2, 1, 1]\n sage: p([2,1]).scalar(q([2,1]))\n 1\n sage: p([2,1]).scalar(q([1,1,1]))\n 0\n "
import dual
if (scalar is None):
if ((basis_name is None) and (prefix is None)):
return self._dual_basis_default()
scalar = zee
scalar_name = 'Hall scalar product'
return dual.SymmetricFunctionAlgebra_dual(self, scalar, scalar_name, basis_name=basis_name, prefix=prefix)
|
Return the dual basis of ``self`` with respect to the scalar
product ``scalar``.
INPUT:
- ``scalar`` -- A function ``zee`` from partitions to the base ring
which specifies the scalar product by `\langle p_{\lambda},
p_{\lambda} \rangle = \mathrm{zee}(\lambda)`. (Independently on the
function chosen, the power sum basis will always be orthogonal;
the function ``scalar`` only determines the norms of the basis
elements.) If ``scalar`` is None, then the standard (Hall) scalar
product is used.
- ``scalar_name`` -- name of the scalar function
- ``prefix`` -- prefix used to display the basis
EXAMPLES:
The duals of the elementary symmetric functions with respect to the
Hall scalar product are the forgotten symmetric functions.
::
sage: e = SymmetricFunctions(QQ).e()
sage: f = e.dual_basis(prefix='f'); f
Dual basis to Symmetric Functions over Rational Field in the elementary basis with respect to the Hall scalar product
sage: f([2,1])^2
4*f[2, 2, 1, 1] + 6*f[2, 2, 2] + 2*f[3, 2, 1] + 2*f[3, 3] + 2*f[4, 1, 1] + f[4, 2]
sage: f([2,1]).scalar(e([2,1]))
1
sage: f([2,1]).scalar(e([1,1,1]))
0
Since the power-sum symmetric functions are orthogonal, their duals
with respect to the Hall scalar product are scalar multiples of
themselves.
::
sage: p = SymmetricFunctions(QQ).p()
sage: q = p.dual_basis(prefix='q'); q
Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product
sage: q([2,1])^2
4*q[2, 2, 1, 1]
sage: p([2,1]).scalar(q([2,1]))
1
sage: p([2,1]).scalar(q([1,1,1]))
0
|
src/sage/combinat/sf/sfa.py
|
dual_basis
|
bopopescu/sagesmc
| 5 |
python
|
def dual_basis(self, scalar=None, scalar_name=, basis_name=None, prefix=None):
"\n Return the dual basis of ``self`` with respect to the scalar\n product ``scalar``.\n\n INPUT:\n\n - ``scalar`` -- A function ``zee`` from partitions to the base ring\n which specifies the scalar product by `\\langle p_{\\lambda},\n p_{\\lambda} \\rangle = \\mathrm{zee}(\\lambda)`. (Independently on the\n function chosen, the power sum basis will always be orthogonal;\n the function ``scalar`` only determines the norms of the basis\n elements.) If ``scalar`` is None, then the standard (Hall) scalar\n product is used.\n - ``scalar_name`` -- name of the scalar function\n - ``prefix`` -- prefix used to display the basis\n\n EXAMPLES:\n\n The duals of the elementary symmetric functions with respect to the\n Hall scalar product are the forgotten symmetric functions.\n\n ::\n\n sage: e = SymmetricFunctions(QQ).e()\n sage: f = e.dual_basis(prefix='f'); f\n Dual basis to Symmetric Functions over Rational Field in the elementary basis with respect to the Hall scalar product\n sage: f([2,1])^2\n 4*f[2, 2, 1, 1] + 6*f[2, 2, 2] + 2*f[3, 2, 1] + 2*f[3, 3] + 2*f[4, 1, 1] + f[4, 2]\n sage: f([2,1]).scalar(e([2,1]))\n 1\n sage: f([2,1]).scalar(e([1,1,1]))\n 0\n\n Since the power-sum symmetric functions are orthogonal, their duals\n with respect to the Hall scalar product are scalar multiples of\n themselves.\n\n ::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: q = p.dual_basis(prefix='q'); q\n Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product\n sage: q([2,1])^2\n 4*q[2, 2, 1, 1]\n sage: p([2,1]).scalar(q([2,1]))\n 1\n sage: p([2,1]).scalar(q([1,1,1]))\n 0\n "
import dual
if (scalar is None):
if ((basis_name is None) and (prefix is None)):
return self._dual_basis_default()
scalar = zee
scalar_name = 'Hall scalar product'
return dual.SymmetricFunctionAlgebra_dual(self, scalar, scalar_name, basis_name=basis_name, prefix=prefix)
|
def dual_basis(self, scalar=None, scalar_name=, basis_name=None, prefix=None):
"\n Return the dual basis of ``self`` with respect to the scalar\n product ``scalar``.\n\n INPUT:\n\n - ``scalar`` -- A function ``zee`` from partitions to the base ring\n which specifies the scalar product by `\\langle p_{\\lambda},\n p_{\\lambda} \\rangle = \\mathrm{zee}(\\lambda)`. (Independently on the\n function chosen, the power sum basis will always be orthogonal;\n the function ``scalar`` only determines the norms of the basis\n elements.) If ``scalar`` is None, then the standard (Hall) scalar\n product is used.\n - ``scalar_name`` -- name of the scalar function\n - ``prefix`` -- prefix used to display the basis\n\n EXAMPLES:\n\n The duals of the elementary symmetric functions with respect to the\n Hall scalar product are the forgotten symmetric functions.\n\n ::\n\n sage: e = SymmetricFunctions(QQ).e()\n sage: f = e.dual_basis(prefix='f'); f\n Dual basis to Symmetric Functions over Rational Field in the elementary basis with respect to the Hall scalar product\n sage: f([2,1])^2\n 4*f[2, 2, 1, 1] + 6*f[2, 2, 2] + 2*f[3, 2, 1] + 2*f[3, 3] + 2*f[4, 1, 1] + f[4, 2]\n sage: f([2,1]).scalar(e([2,1]))\n 1\n sage: f([2,1]).scalar(e([1,1,1]))\n 0\n\n Since the power-sum symmetric functions are orthogonal, their duals\n with respect to the Hall scalar product are scalar multiples of\n themselves.\n\n ::\n\n sage: p = SymmetricFunctions(QQ).p()\n sage: q = p.dual_basis(prefix='q'); q\n Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product\n sage: q([2,1])^2\n 4*q[2, 2, 1, 1]\n sage: p([2,1]).scalar(q([2,1]))\n 1\n sage: p([2,1]).scalar(q([1,1,1]))\n 0\n "
import dual
if (scalar is None):
if ((basis_name is None) and (prefix is None)):
return self._dual_basis_default()
scalar = zee
scalar_name = 'Hall scalar product'
return dual.SymmetricFunctionAlgebra_dual(self, scalar, scalar_name, basis_name=basis_name, prefix=prefix)<|docstring|>Return the dual basis of ``self`` with respect to the scalar
product ``scalar``.
INPUT:
- ``scalar`` -- A function ``zee`` from partitions to the base ring
which specifies the scalar product by `\langle p_{\lambda},
p_{\lambda} \rangle = \mathrm{zee}(\lambda)`. (Independently on the
function chosen, the power sum basis will always be orthogonal;
the function ``scalar`` only determines the norms of the basis
elements.) If ``scalar`` is None, then the standard (Hall) scalar
product is used.
- ``scalar_name`` -- name of the scalar function
- ``prefix`` -- prefix used to display the basis
EXAMPLES:
The duals of the elementary symmetric functions with respect to the
Hall scalar product are the forgotten symmetric functions.
::
sage: e = SymmetricFunctions(QQ).e()
sage: f = e.dual_basis(prefix='f'); f
Dual basis to Symmetric Functions over Rational Field in the elementary basis with respect to the Hall scalar product
sage: f([2,1])^2
4*f[2, 2, 1, 1] + 6*f[2, 2, 2] + 2*f[3, 2, 1] + 2*f[3, 3] + 2*f[4, 1, 1] + f[4, 2]
sage: f([2,1]).scalar(e([2,1]))
1
sage: f([2,1]).scalar(e([1,1,1]))
0
Since the power-sum symmetric functions are orthogonal, their duals
with respect to the Hall scalar product are scalar multiples of
themselves.
::
sage: p = SymmetricFunctions(QQ).p()
sage: q = p.dual_basis(prefix='q'); q
Dual basis to Symmetric Functions over Rational Field in the powersum basis with respect to the Hall scalar product
sage: q([2,1])^2
4*q[2, 2, 1, 1]
sage: p([2,1]).scalar(q([2,1]))
1
sage: p([2,1]).scalar(q([1,1,1]))
0<|endoftext|>
|
c87298bbb07b02e31aaf84b9cdeb3e4cb5fd520fbe2ed0ab35a653ce15001eea
|
def basis_name(self):
"\n Return the name of the basis of ``self``.\n\n This is used for output and, for the classical bases of\n symmetric functions, to connect this basis with Symmetrica.\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: s = Sym.s()\n sage: s.basis_name()\n 'Schur'\n sage: p = Sym.p()\n sage: p.basis_name()\n 'powersum'\n sage: h = Sym.h()\n sage: h.basis_name()\n 'homogeneous'\n sage: e = Sym.e()\n sage: e.basis_name()\n 'elementary'\n sage: m = Sym.m()\n sage: m.basis_name()\n 'monomial'\n sage: f = Sym.f()\n sage: f.basis_name()\n 'forgotten'\n "
return self._basis
|
Return the name of the basis of ``self``.
This is used for output and, for the classical bases of
symmetric functions, to connect this basis with Symmetrica.
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.s()
sage: s.basis_name()
'Schur'
sage: p = Sym.p()
sage: p.basis_name()
'powersum'
sage: h = Sym.h()
sage: h.basis_name()
'homogeneous'
sage: e = Sym.e()
sage: e.basis_name()
'elementary'
sage: m = Sym.m()
sage: m.basis_name()
'monomial'
sage: f = Sym.f()
sage: f.basis_name()
'forgotten'
|
src/sage/combinat/sf/sfa.py
|
basis_name
|
bopopescu/sagesmc
| 5 |
python
|
def basis_name(self):
"\n Return the name of the basis of ``self``.\n\n This is used for output and, for the classical bases of\n symmetric functions, to connect this basis with Symmetrica.\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: s = Sym.s()\n sage: s.basis_name()\n 'Schur'\n sage: p = Sym.p()\n sage: p.basis_name()\n 'powersum'\n sage: h = Sym.h()\n sage: h.basis_name()\n 'homogeneous'\n sage: e = Sym.e()\n sage: e.basis_name()\n 'elementary'\n sage: m = Sym.m()\n sage: m.basis_name()\n 'monomial'\n sage: f = Sym.f()\n sage: f.basis_name()\n 'forgotten'\n "
return self._basis
|
def basis_name(self):
"\n Return the name of the basis of ``self``.\n\n This is used for output and, for the classical bases of\n symmetric functions, to connect this basis with Symmetrica.\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: s = Sym.s()\n sage: s.basis_name()\n 'Schur'\n sage: p = Sym.p()\n sage: p.basis_name()\n 'powersum'\n sage: h = Sym.h()\n sage: h.basis_name()\n 'homogeneous'\n sage: e = Sym.e()\n sage: e.basis_name()\n 'elementary'\n sage: m = Sym.m()\n sage: m.basis_name()\n 'monomial'\n sage: f = Sym.f()\n sage: f.basis_name()\n 'forgotten'\n "
return self._basis<|docstring|>Return the name of the basis of ``self``.
This is used for output and, for the classical bases of
symmetric functions, to connect this basis with Symmetrica.
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.s()
sage: s.basis_name()
'Schur'
sage: p = Sym.p()
sage: p.basis_name()
'powersum'
sage: h = Sym.h()
sage: h.basis_name()
'homogeneous'
sage: e = Sym.e()
sage: e.basis_name()
'elementary'
sage: m = Sym.m()
sage: m.basis_name()
'monomial'
sage: f = Sym.f()
sage: f.basis_name()
'forgotten'<|endoftext|>
|
9026b4db0ddd47efa022da7e6762efaa9c3c9d87bc71b3d161c4387adddf4969
|
def get_print_style(self):
"\n Return the value of the current print style for ``self``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s.get_print_style()\n 'lex'\n sage: s.set_print_style('length')\n sage: s.get_print_style()\n 'length'\n sage: s.set_print_style('lex')\n "
return self._print_style
|
Return the value of the current print style for ``self``.
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: s.get_print_style()
'lex'
sage: s.set_print_style('length')
sage: s.get_print_style()
'length'
sage: s.set_print_style('lex')
|
src/sage/combinat/sf/sfa.py
|
get_print_style
|
bopopescu/sagesmc
| 5 |
python
|
def get_print_style(self):
"\n Return the value of the current print style for ``self``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s.get_print_style()\n 'lex'\n sage: s.set_print_style('length')\n sage: s.get_print_style()\n 'length'\n sage: s.set_print_style('lex')\n "
return self._print_style
|
def get_print_style(self):
"\n Return the value of the current print style for ``self``.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s.get_print_style()\n 'lex'\n sage: s.set_print_style('length')\n sage: s.get_print_style()\n 'length'\n sage: s.set_print_style('lex')\n "
return self._print_style<|docstring|>Return the value of the current print style for ``self``.
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: s.get_print_style()
'lex'
sage: s.set_print_style('length')
sage: s.get_print_style()
'length'
sage: s.set_print_style('lex')<|endoftext|>
|
d8df14daa008c4667e786254ca5f7e91d1dc72e86709e3b0454c551a92927c54
|
def set_print_style(self, ps):
"\n Set the value of the current print style to ``ps``.\n\n INPUT:\n\n - ``ps`` -- a string specifying the printing style\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s.get_print_style()\n 'lex'\n sage: s.set_print_style('length')\n sage: s.get_print_style()\n 'length'\n sage: s.set_print_style('lex')\n "
if (ps == 'lex'):
self.print_options(monomial_cmp=(lambda x, y: cmp(x, y)))
elif (ps == 'length'):
self.print_options(monomial_cmp=(lambda x, y: cmp(len(x), len(y))))
elif (ps == 'maximal_part'):
self.print_options(monomial_cmp=(lambda x, y: cmp(_lmax(x), _lmax(y))))
else:
raise ValueError('the print style must be one of lex, length, or maximal_part ')
self._print_style = ps
|
Set the value of the current print style to ``ps``.
INPUT:
- ``ps`` -- a string specifying the printing style
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: s.get_print_style()
'lex'
sage: s.set_print_style('length')
sage: s.get_print_style()
'length'
sage: s.set_print_style('lex')
|
src/sage/combinat/sf/sfa.py
|
set_print_style
|
bopopescu/sagesmc
| 5 |
python
|
def set_print_style(self, ps):
"\n Set the value of the current print style to ``ps``.\n\n INPUT:\n\n - ``ps`` -- a string specifying the printing style\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s.get_print_style()\n 'lex'\n sage: s.set_print_style('length')\n sage: s.get_print_style()\n 'length'\n sage: s.set_print_style('lex')\n "
if (ps == 'lex'):
self.print_options(monomial_cmp=(lambda x, y: cmp(x, y)))
elif (ps == 'length'):
self.print_options(monomial_cmp=(lambda x, y: cmp(len(x), len(y))))
elif (ps == 'maximal_part'):
self.print_options(monomial_cmp=(lambda x, y: cmp(_lmax(x), _lmax(y))))
else:
raise ValueError('the print style must be one of lex, length, or maximal_part ')
self._print_style = ps
|
def set_print_style(self, ps):
"\n Set the value of the current print style to ``ps``.\n\n INPUT:\n\n - ``ps`` -- a string specifying the printing style\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s.get_print_style()\n 'lex'\n sage: s.set_print_style('length')\n sage: s.get_print_style()\n 'length'\n sage: s.set_print_style('lex')\n "
if (ps == 'lex'):
self.print_options(monomial_cmp=(lambda x, y: cmp(x, y)))
elif (ps == 'length'):
self.print_options(monomial_cmp=(lambda x, y: cmp(len(x), len(y))))
elif (ps == 'maximal_part'):
self.print_options(monomial_cmp=(lambda x, y: cmp(_lmax(x), _lmax(y))))
else:
raise ValueError('the print style must be one of lex, length, or maximal_part ')
self._print_style = ps<|docstring|>Set the value of the current print style to ``ps``.
INPUT:
- ``ps`` -- a string specifying the printing style
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: s.get_print_style()
'lex'
sage: s.set_print_style('length')
sage: s.get_print_style()
'length'
sage: s.set_print_style('lex')<|endoftext|>
|
b2146b7041f5e6d5ec575d88854ba9b3c74a031a98a7e528be9ccd6be15f31eb
|
def _latex_term(self, m):
"\n Latex terms (i.e. partitions) as plain lists (and not as\n ferrers diagrams).\n\n INPUT:\n\n - ``m`` -- a partition or list\n\n EXAMPLES::\n\n sage: m = SymmetricFunctions(QQ).m()\n sage: m._latex_term(Partition([3,2,1]))\n 'm_{3,2,1}'\n sage: f = sum([m(p) for p in Partitions(3)])\n sage: m.set_print_style('lex')\n sage: latex(f)\n m_{1,1,1} + m_{2,1} + m_{3}\n sage: m.set_print_style('length')\n sage: latex(f)\n m_{3} + m_{2,1} + m_{1,1,1}\n sage: m.set_print_style('maximal_part')\n sage: latex(f)\n m_{1,1,1} + m_{2,1} + m_{3}\n "
return super(SymmetricFunctionAlgebra_generic, self)._latex_term(','.join((str(i) for i in m)))
|
Latex terms (i.e. partitions) as plain lists (and not as
ferrers diagrams).
INPUT:
- ``m`` -- a partition or list
EXAMPLES::
sage: m = SymmetricFunctions(QQ).m()
sage: m._latex_term(Partition([3,2,1]))
'm_{3,2,1}'
sage: f = sum([m(p) for p in Partitions(3)])
sage: m.set_print_style('lex')
sage: latex(f)
m_{1,1,1} + m_{2,1} + m_{3}
sage: m.set_print_style('length')
sage: latex(f)
m_{3} + m_{2,1} + m_{1,1,1}
sage: m.set_print_style('maximal_part')
sage: latex(f)
m_{1,1,1} + m_{2,1} + m_{3}
|
src/sage/combinat/sf/sfa.py
|
_latex_term
|
bopopescu/sagesmc
| 5 |
python
|
def _latex_term(self, m):
"\n Latex terms (i.e. partitions) as plain lists (and not as\n ferrers diagrams).\n\n INPUT:\n\n - ``m`` -- a partition or list\n\n EXAMPLES::\n\n sage: m = SymmetricFunctions(QQ).m()\n sage: m._latex_term(Partition([3,2,1]))\n 'm_{3,2,1}'\n sage: f = sum([m(p) for p in Partitions(3)])\n sage: m.set_print_style('lex')\n sage: latex(f)\n m_{1,1,1} + m_{2,1} + m_{3}\n sage: m.set_print_style('length')\n sage: latex(f)\n m_{3} + m_{2,1} + m_{1,1,1}\n sage: m.set_print_style('maximal_part')\n sage: latex(f)\n m_{1,1,1} + m_{2,1} + m_{3}\n "
return super(SymmetricFunctionAlgebra_generic, self)._latex_term(','.join((str(i) for i in m)))
|
def _latex_term(self, m):
"\n Latex terms (i.e. partitions) as plain lists (and not as\n ferrers diagrams).\n\n INPUT:\n\n - ``m`` -- a partition or list\n\n EXAMPLES::\n\n sage: m = SymmetricFunctions(QQ).m()\n sage: m._latex_term(Partition([3,2,1]))\n 'm_{3,2,1}'\n sage: f = sum([m(p) for p in Partitions(3)])\n sage: m.set_print_style('lex')\n sage: latex(f)\n m_{1,1,1} + m_{2,1} + m_{3}\n sage: m.set_print_style('length')\n sage: latex(f)\n m_{3} + m_{2,1} + m_{1,1,1}\n sage: m.set_print_style('maximal_part')\n sage: latex(f)\n m_{1,1,1} + m_{2,1} + m_{3}\n "
return super(SymmetricFunctionAlgebra_generic, self)._latex_term(','.join((str(i) for i in m)))<|docstring|>Latex terms (i.e. partitions) as plain lists (and not as
ferrers diagrams).
INPUT:
- ``m`` -- a partition or list
EXAMPLES::
sage: m = SymmetricFunctions(QQ).m()
sage: m._latex_term(Partition([3,2,1]))
'm_{3,2,1}'
sage: f = sum([m(p) for p in Partitions(3)])
sage: m.set_print_style('lex')
sage: latex(f)
m_{1,1,1} + m_{2,1} + m_{3}
sage: m.set_print_style('length')
sage: latex(f)
m_{3} + m_{2,1} + m_{1,1,1}
sage: m.set_print_style('maximal_part')
sage: latex(f)
m_{1,1,1} + m_{2,1} + m_{3}<|endoftext|>
|
8a760f397a206b9d69c3696c8809ee733f275847321b23a63f7cc2ceb22e3cbb
|
def from_polynomial(self, poly, check=True):
'\n Convert polynomial to a symmetric function in the monomial basis\n and then to the basis ``self``.\n\n INPUT:\n\n - ``poly`` -- a symmetric polynomial\n - ``check`` -- (default: ``True``) boolean, specifies whether\n the computation checks that the polynomial is indeed symmetric\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: h = Sym.homogeneous()\n sage: f = (h([]) + h([2,1]) + h([3])).expand(3)\n sage: h.from_polynomial(f)\n h[] + h[2, 1] + h[3]\n sage: s = Sym.s()\n sage: g = (s([]) + s([2,1])).expand(3); g\n x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 + 1\n sage: s.from_polynomial(g)\n s[] + s[2, 1]\n '
m = self.realization_of().m()
return self(m.from_polynomial(poly, check=check))
|
Convert polynomial to a symmetric function in the monomial basis
and then to the basis ``self``.
INPUT:
- ``poly`` -- a symmetric polynomial
- ``check`` -- (default: ``True``) boolean, specifies whether
the computation checks that the polynomial is indeed symmetric
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ)
sage: h = Sym.homogeneous()
sage: f = (h([]) + h([2,1]) + h([3])).expand(3)
sage: h.from_polynomial(f)
h[] + h[2, 1] + h[3]
sage: s = Sym.s()
sage: g = (s([]) + s([2,1])).expand(3); g
x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 + 1
sage: s.from_polynomial(g)
s[] + s[2, 1]
|
src/sage/combinat/sf/sfa.py
|
from_polynomial
|
bopopescu/sagesmc
| 5 |
python
|
def from_polynomial(self, poly, check=True):
'\n Convert polynomial to a symmetric function in the monomial basis\n and then to the basis ``self``.\n\n INPUT:\n\n - ``poly`` -- a symmetric polynomial\n - ``check`` -- (default: ``True``) boolean, specifies whether\n the computation checks that the polynomial is indeed symmetric\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: h = Sym.homogeneous()\n sage: f = (h([]) + h([2,1]) + h([3])).expand(3)\n sage: h.from_polynomial(f)\n h[] + h[2, 1] + h[3]\n sage: s = Sym.s()\n sage: g = (s([]) + s([2,1])).expand(3); g\n x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 + 1\n sage: s.from_polynomial(g)\n s[] + s[2, 1]\n '
m = self.realization_of().m()
return self(m.from_polynomial(poly, check=check))
|
def from_polynomial(self, poly, check=True):
'\n Convert polynomial to a symmetric function in the monomial basis\n and then to the basis ``self``.\n\n INPUT:\n\n - ``poly`` -- a symmetric polynomial\n - ``check`` -- (default: ``True``) boolean, specifies whether\n the computation checks that the polynomial is indeed symmetric\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: h = Sym.homogeneous()\n sage: f = (h([]) + h([2,1]) + h([3])).expand(3)\n sage: h.from_polynomial(f)\n h[] + h[2, 1] + h[3]\n sage: s = Sym.s()\n sage: g = (s([]) + s([2,1])).expand(3); g\n x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 + 1\n sage: s.from_polynomial(g)\n s[] + s[2, 1]\n '
m = self.realization_of().m()
return self(m.from_polynomial(poly, check=check))<|docstring|>Convert polynomial to a symmetric function in the monomial basis
and then to the basis ``self``.
INPUT:
- ``poly`` -- a symmetric polynomial
- ``check`` -- (default: ``True``) boolean, specifies whether
the computation checks that the polynomial is indeed symmetric
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ)
sage: h = Sym.homogeneous()
sage: f = (h([]) + h([2,1]) + h([3])).expand(3)
sage: h.from_polynomial(f)
h[] + h[2, 1] + h[3]
sage: s = Sym.s()
sage: g = (s([]) + s([2,1])).expand(3); g
x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 + 1
sage: s.from_polynomial(g)
s[] + s[2, 1]<|endoftext|>
|
2443f83a1d1d37af9bf86499257ba6afa250775346a13ddbfc0f203a21809052
|
def coproduct_by_coercion(self, elt):
"\n Return the coproduct of the element ``elt`` by coercion to\n the Schur basis.\n\n INPUT:\n\n - ``elt`` -- an instance of this basis\n\n OUTPUT:\n\n - The coproduct acting on ``elt``, the result is an element of the\n tensor squared of the basis ``self``\n\n EXAMPLES::\n\n sage: m = SymmetricFunctions(QQ).m()\n sage: m[3,1,1].coproduct()\n m[] # m[3, 1, 1] + m[1] # m[3, 1] + m[1, 1] # m[3] + m[3] # m[1, 1] + m[3, 1] # m[1] + m[3, 1, 1] # m[]\n sage: m.coproduct_by_coercion(m[2,1])\n m[] # m[2, 1] + m[1] # m[2] + m[2] # m[1] + m[2, 1] # m[]\n sage: m.coproduct_by_coercion(m[2,1]) == m([2,1]).coproduct()\n True\n sage: McdH = SymmetricFunctions(QQ['q','t'].fraction_field()).macdonald().H()\n sage: McdH[2,1].coproduct()\n McdH[] # McdH[2, 1] + ((q^2*t-1)/(q*t-1))*McdH[1] # McdH[1, 1] + ((q*t^2-1)/(q*t-1))*McdH[1] # McdH[2] + ((q^2*t-1)/(q*t-1))*McdH[1, 1] # McdH[1] + ((q*t^2-1)/(q*t-1))*McdH[2] # McdH[1] + McdH[2, 1] # McdH[]\n sage: HLQp = SymmetricFunctions(QQ['t'].fraction_field()).hall_littlewood().Qp()\n sage: HLQp[2,1].coproduct()\n HLQp[] # HLQp[2, 1] + HLQp[1] # HLQp[1, 1] + HLQp[1] # HLQp[2] + HLQp[1, 1] # HLQp[1] + HLQp[2] # HLQp[1] + HLQp[2, 1] # HLQp[]\n sage: Sym = SymmetricFunctions(FractionField(QQ['t']))\n sage: LLT = Sym.llt(3)\n sage: LLT.cospin([3,2,1]).coproduct()\n (t+1)*m[] # m[1, 1] + m[] # m[2] + (t+1)*m[1] # m[1] + (t+1)*m[1, 1] # m[] + m[2] # m[]\n sage: f = SymmetricFunctions(ZZ).f()\n sage: f[3].coproduct()\n f[] # f[3] + f[3] # f[]\n sage: f[3,2,1].coproduct()\n f[] # f[3, 2, 1] + f[1] # f[3, 2] + f[2] # f[3, 1] + f[2, 1] # f[3] + f[3] # f[2, 1] + f[3, 1] # f[2] + f[3, 2] # f[1] + f[3, 2, 1] # f[]\n "
from sage.categories.tensor import tensor
s = self.realization_of().schur()
return self.tensor_square().sum(((coeff * tensor([self(s[x]), self(s[y])])) for ((x, y), coeff) in s(elt).coproduct()))
|
Return the coproduct of the element ``elt`` by coercion to
the Schur basis.
INPUT:
- ``elt`` -- an instance of this basis
OUTPUT:
- The coproduct acting on ``elt``, the result is an element of the
tensor squared of the basis ``self``
EXAMPLES::
sage: m = SymmetricFunctions(QQ).m()
sage: m[3,1,1].coproduct()
m[] # m[3, 1, 1] + m[1] # m[3, 1] + m[1, 1] # m[3] + m[3] # m[1, 1] + m[3, 1] # m[1] + m[3, 1, 1] # m[]
sage: m.coproduct_by_coercion(m[2,1])
m[] # m[2, 1] + m[1] # m[2] + m[2] # m[1] + m[2, 1] # m[]
sage: m.coproduct_by_coercion(m[2,1]) == m([2,1]).coproduct()
True
sage: McdH = SymmetricFunctions(QQ['q','t'].fraction_field()).macdonald().H()
sage: McdH[2,1].coproduct()
McdH[] # McdH[2, 1] + ((q^2*t-1)/(q*t-1))*McdH[1] # McdH[1, 1] + ((q*t^2-1)/(q*t-1))*McdH[1] # McdH[2] + ((q^2*t-1)/(q*t-1))*McdH[1, 1] # McdH[1] + ((q*t^2-1)/(q*t-1))*McdH[2] # McdH[1] + McdH[2, 1] # McdH[]
sage: HLQp = SymmetricFunctions(QQ['t'].fraction_field()).hall_littlewood().Qp()
sage: HLQp[2,1].coproduct()
HLQp[] # HLQp[2, 1] + HLQp[1] # HLQp[1, 1] + HLQp[1] # HLQp[2] + HLQp[1, 1] # HLQp[1] + HLQp[2] # HLQp[1] + HLQp[2, 1] # HLQp[]
sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: LLT = Sym.llt(3)
sage: LLT.cospin([3,2,1]).coproduct()
(t+1)*m[] # m[1, 1] + m[] # m[2] + (t+1)*m[1] # m[1] + (t+1)*m[1, 1] # m[] + m[2] # m[]
sage: f = SymmetricFunctions(ZZ).f()
sage: f[3].coproduct()
f[] # f[3] + f[3] # f[]
sage: f[3,2,1].coproduct()
f[] # f[3, 2, 1] + f[1] # f[3, 2] + f[2] # f[3, 1] + f[2, 1] # f[3] + f[3] # f[2, 1] + f[3, 1] # f[2] + f[3, 2] # f[1] + f[3, 2, 1] # f[]
|
src/sage/combinat/sf/sfa.py
|
coproduct_by_coercion
|
bopopescu/sagesmc
| 5 |
python
|
def coproduct_by_coercion(self, elt):
"\n Return the coproduct of the element ``elt`` by coercion to\n the Schur basis.\n\n INPUT:\n\n - ``elt`` -- an instance of this basis\n\n OUTPUT:\n\n - The coproduct acting on ``elt``, the result is an element of the\n tensor squared of the basis ``self``\n\n EXAMPLES::\n\n sage: m = SymmetricFunctions(QQ).m()\n sage: m[3,1,1].coproduct()\n m[] # m[3, 1, 1] + m[1] # m[3, 1] + m[1, 1] # m[3] + m[3] # m[1, 1] + m[3, 1] # m[1] + m[3, 1, 1] # m[]\n sage: m.coproduct_by_coercion(m[2,1])\n m[] # m[2, 1] + m[1] # m[2] + m[2] # m[1] + m[2, 1] # m[]\n sage: m.coproduct_by_coercion(m[2,1]) == m([2,1]).coproduct()\n True\n sage: McdH = SymmetricFunctions(QQ['q','t'].fraction_field()).macdonald().H()\n sage: McdH[2,1].coproduct()\n McdH[] # McdH[2, 1] + ((q^2*t-1)/(q*t-1))*McdH[1] # McdH[1, 1] + ((q*t^2-1)/(q*t-1))*McdH[1] # McdH[2] + ((q^2*t-1)/(q*t-1))*McdH[1, 1] # McdH[1] + ((q*t^2-1)/(q*t-1))*McdH[2] # McdH[1] + McdH[2, 1] # McdH[]\n sage: HLQp = SymmetricFunctions(QQ['t'].fraction_field()).hall_littlewood().Qp()\n sage: HLQp[2,1].coproduct()\n HLQp[] # HLQp[2, 1] + HLQp[1] # HLQp[1, 1] + HLQp[1] # HLQp[2] + HLQp[1, 1] # HLQp[1] + HLQp[2] # HLQp[1] + HLQp[2, 1] # HLQp[]\n sage: Sym = SymmetricFunctions(FractionField(QQ['t']))\n sage: LLT = Sym.llt(3)\n sage: LLT.cospin([3,2,1]).coproduct()\n (t+1)*m[] # m[1, 1] + m[] # m[2] + (t+1)*m[1] # m[1] + (t+1)*m[1, 1] # m[] + m[2] # m[]\n sage: f = SymmetricFunctions(ZZ).f()\n sage: f[3].coproduct()\n f[] # f[3] + f[3] # f[]\n sage: f[3,2,1].coproduct()\n f[] # f[3, 2, 1] + f[1] # f[3, 2] + f[2] # f[3, 1] + f[2, 1] # f[3] + f[3] # f[2, 1] + f[3, 1] # f[2] + f[3, 2] # f[1] + f[3, 2, 1] # f[]\n "
from sage.categories.tensor import tensor
s = self.realization_of().schur()
return self.tensor_square().sum(((coeff * tensor([self(s[x]), self(s[y])])) for ((x, y), coeff) in s(elt).coproduct()))
|
def coproduct_by_coercion(self, elt):
"\n Return the coproduct of the element ``elt`` by coercion to\n the Schur basis.\n\n INPUT:\n\n - ``elt`` -- an instance of this basis\n\n OUTPUT:\n\n - The coproduct acting on ``elt``, the result is an element of the\n tensor squared of the basis ``self``\n\n EXAMPLES::\n\n sage: m = SymmetricFunctions(QQ).m()\n sage: m[3,1,1].coproduct()\n m[] # m[3, 1, 1] + m[1] # m[3, 1] + m[1, 1] # m[3] + m[3] # m[1, 1] + m[3, 1] # m[1] + m[3, 1, 1] # m[]\n sage: m.coproduct_by_coercion(m[2,1])\n m[] # m[2, 1] + m[1] # m[2] + m[2] # m[1] + m[2, 1] # m[]\n sage: m.coproduct_by_coercion(m[2,1]) == m([2,1]).coproduct()\n True\n sage: McdH = SymmetricFunctions(QQ['q','t'].fraction_field()).macdonald().H()\n sage: McdH[2,1].coproduct()\n McdH[] # McdH[2, 1] + ((q^2*t-1)/(q*t-1))*McdH[1] # McdH[1, 1] + ((q*t^2-1)/(q*t-1))*McdH[1] # McdH[2] + ((q^2*t-1)/(q*t-1))*McdH[1, 1] # McdH[1] + ((q*t^2-1)/(q*t-1))*McdH[2] # McdH[1] + McdH[2, 1] # McdH[]\n sage: HLQp = SymmetricFunctions(QQ['t'].fraction_field()).hall_littlewood().Qp()\n sage: HLQp[2,1].coproduct()\n HLQp[] # HLQp[2, 1] + HLQp[1] # HLQp[1, 1] + HLQp[1] # HLQp[2] + HLQp[1, 1] # HLQp[1] + HLQp[2] # HLQp[1] + HLQp[2, 1] # HLQp[]\n sage: Sym = SymmetricFunctions(FractionField(QQ['t']))\n sage: LLT = Sym.llt(3)\n sage: LLT.cospin([3,2,1]).coproduct()\n (t+1)*m[] # m[1, 1] + m[] # m[2] + (t+1)*m[1] # m[1] + (t+1)*m[1, 1] # m[] + m[2] # m[]\n sage: f = SymmetricFunctions(ZZ).f()\n sage: f[3].coproduct()\n f[] # f[3] + f[3] # f[]\n sage: f[3,2,1].coproduct()\n f[] # f[3, 2, 1] + f[1] # f[3, 2] + f[2] # f[3, 1] + f[2, 1] # f[3] + f[3] # f[2, 1] + f[3, 1] # f[2] + f[3, 2] # f[1] + f[3, 2, 1] # f[]\n "
from sage.categories.tensor import tensor
s = self.realization_of().schur()
return self.tensor_square().sum(((coeff * tensor([self(s[x]), self(s[y])])) for ((x, y), coeff) in s(elt).coproduct()))<|docstring|>Return the coproduct of the element ``elt`` by coercion to
the Schur basis.
INPUT:
- ``elt`` -- an instance of this basis
OUTPUT:
- The coproduct acting on ``elt``, the result is an element of the
tensor squared of the basis ``self``
EXAMPLES::
sage: m = SymmetricFunctions(QQ).m()
sage: m[3,1,1].coproduct()
m[] # m[3, 1, 1] + m[1] # m[3, 1] + m[1, 1] # m[3] + m[3] # m[1, 1] + m[3, 1] # m[1] + m[3, 1, 1] # m[]
sage: m.coproduct_by_coercion(m[2,1])
m[] # m[2, 1] + m[1] # m[2] + m[2] # m[1] + m[2, 1] # m[]
sage: m.coproduct_by_coercion(m[2,1]) == m([2,1]).coproduct()
True
sage: McdH = SymmetricFunctions(QQ['q','t'].fraction_field()).macdonald().H()
sage: McdH[2,1].coproduct()
McdH[] # McdH[2, 1] + ((q^2*t-1)/(q*t-1))*McdH[1] # McdH[1, 1] + ((q*t^2-1)/(q*t-1))*McdH[1] # McdH[2] + ((q^2*t-1)/(q*t-1))*McdH[1, 1] # McdH[1] + ((q*t^2-1)/(q*t-1))*McdH[2] # McdH[1] + McdH[2, 1] # McdH[]
sage: HLQp = SymmetricFunctions(QQ['t'].fraction_field()).hall_littlewood().Qp()
sage: HLQp[2,1].coproduct()
HLQp[] # HLQp[2, 1] + HLQp[1] # HLQp[1, 1] + HLQp[1] # HLQp[2] + HLQp[1, 1] # HLQp[1] + HLQp[2] # HLQp[1] + HLQp[2, 1] # HLQp[]
sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: LLT = Sym.llt(3)
sage: LLT.cospin([3,2,1]).coproduct()
(t+1)*m[] # m[1, 1] + m[] # m[2] + (t+1)*m[1] # m[1] + (t+1)*m[1, 1] # m[] + m[2] # m[]
sage: f = SymmetricFunctions(ZZ).f()
sage: f[3].coproduct()
f[] # f[3] + f[3] # f[]
sage: f[3,2,1].coproduct()
f[] # f[3, 2, 1] + f[1] # f[3, 2] + f[2] # f[3, 1] + f[2, 1] # f[3] + f[3] # f[2, 1] + f[3, 1] # f[2] + f[3, 2] # f[1] + f[3, 2, 1] # f[]<|endoftext|>
|
be47ff922606f3fbd460d197c2527a1759d45fe42b0b077f86d70f9273c66c04
|
def plethysm(self, x, include=None, exclude=None):
"\n Return the outer plethysm of ``self`` with ``x``. This is\n implemented only over base rings which are `\\QQ`-algebras.\n (To compute outer plethysms over general binomial rings, change\n bases to the fraction field.)\n\n By default, the degree one elements are taken to be the\n generators for the ``self``'s base ring. This setting can be\n modified by specifying the ``include`` and ``exclude`` keywords.\n\n INPUT:\n\n - ``x`` -- a symmetric function over the same base ring as\n ``self``\n\n - ``include`` -- a list of variables to be treated as\n degree one elements instead of the default degree one elements\n\n - ``exclude`` -- a list of variables to be excluded\n from the default degree one elements\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: s = Sym.s()\n sage: h = Sym.h()\n sage: s ( h([3])( h([2]) ) )\n s[2, 2, 2] + s[4, 2] + s[6]\n sage: p = Sym.p()\n sage: p([3])( s([2,1]) )\n 1/3*p[3, 3, 3] - 1/3*p[9]\n sage: e = Sym.e()\n sage: e([3])( e([2]) )\n e[3, 3] + e[4, 1, 1] - 2*e[4, 2] - e[5, 1] + e[6]\n\n ::\n\n sage: R.<t> = QQ[]\n sage: s = SymmetricFunctions(R).s()\n sage: a = s([3])\n sage: f = t*s([2])\n sage: a(f)\n t^3*s[2, 2, 2] + t^3*s[4, 2] + t^3*s[6]\n sage: f(a)\n t*s[4, 2] + t*s[6]\n sage: s(0).plethysm(s[1])\n 0\n sage: s(1).plethysm(s[1])\n s[]\n sage: s(1).plethysm(s(0))\n s[]\n\n .. SEEALSO::\n\n :meth:`frobenius`\n "
if (not is_SymmetricFunction(x)):
raise TypeError('only know how to compute plethysms between symmetric functions')
parent = self.parent()
p = parent.realization_of().power()
R = parent.base_ring()
p_x = p(x)
if (self == parent.zero()):
return self
if ((include is not None) and (exclude is not None)):
raise RuntimeError('include and exclude cannot both be specified')
try:
degree_one = [R(g) for g in R.variable_names_recursive()]
except AttributeError:
try:
degree_one = R.gens()
except NotImplementedError:
degree_one = []
if include:
degree_one = [R(g) for g in include]
if exclude:
degree_one = [g for g in degree_one if (g not in exclude)]
scale_part = (lambda n: (lambda m: m.__class__(m.parent(), [(i * n) for i in m])))
raise_c = (lambda n: (lambda c: c.subs(**dict(((str(g), (g ** n)) for g in degree_one if (g != 1))))))
pn_pleth = (lambda f, n: f.map_support(scale_part(n)))
f = (lambda part: prod((pn_pleth(p_x.map_coefficients(raise_c(i)), i) for i in part)))
return parent(p._apply_module_morphism(p(self), f))
|
Return the outer plethysm of ``self`` with ``x``. This is
implemented only over base rings which are `\QQ`-algebras.
(To compute outer plethysms over general binomial rings, change
bases to the fraction field.)
By default, the degree one elements are taken to be the
generators for the ``self``'s base ring. This setting can be
modified by specifying the ``include`` and ``exclude`` keywords.
INPUT:
- ``x`` -- a symmetric function over the same base ring as
``self``
- ``include`` -- a list of variables to be treated as
degree one elements instead of the default degree one elements
- ``exclude`` -- a list of variables to be excluded
from the default degree one elements
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.s()
sage: h = Sym.h()
sage: s ( h([3])( h([2]) ) )
s[2, 2, 2] + s[4, 2] + s[6]
sage: p = Sym.p()
sage: p([3])( s([2,1]) )
1/3*p[3, 3, 3] - 1/3*p[9]
sage: e = Sym.e()
sage: e([3])( e([2]) )
e[3, 3] + e[4, 1, 1] - 2*e[4, 2] - e[5, 1] + e[6]
::
sage: R.<t> = QQ[]
sage: s = SymmetricFunctions(R).s()
sage: a = s([3])
sage: f = t*s([2])
sage: a(f)
t^3*s[2, 2, 2] + t^3*s[4, 2] + t^3*s[6]
sage: f(a)
t*s[4, 2] + t*s[6]
sage: s(0).plethysm(s[1])
0
sage: s(1).plethysm(s[1])
s[]
sage: s(1).plethysm(s(0))
s[]
.. SEEALSO::
:meth:`frobenius`
|
src/sage/combinat/sf/sfa.py
|
plethysm
|
bopopescu/sagesmc
| 5 |
python
|
def plethysm(self, x, include=None, exclude=None):
"\n Return the outer plethysm of ``self`` with ``x``. This is\n implemented only over base rings which are `\\QQ`-algebras.\n (To compute outer plethysms over general binomial rings, change\n bases to the fraction field.)\n\n By default, the degree one elements are taken to be the\n generators for the ``self``'s base ring. This setting can be\n modified by specifying the ``include`` and ``exclude`` keywords.\n\n INPUT:\n\n - ``x`` -- a symmetric function over the same base ring as\n ``self``\n\n - ``include`` -- a list of variables to be treated as\n degree one elements instead of the default degree one elements\n\n - ``exclude`` -- a list of variables to be excluded\n from the default degree one elements\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: s = Sym.s()\n sage: h = Sym.h()\n sage: s ( h([3])( h([2]) ) )\n s[2, 2, 2] + s[4, 2] + s[6]\n sage: p = Sym.p()\n sage: p([3])( s([2,1]) )\n 1/3*p[3, 3, 3] - 1/3*p[9]\n sage: e = Sym.e()\n sage: e([3])( e([2]) )\n e[3, 3] + e[4, 1, 1] - 2*e[4, 2] - e[5, 1] + e[6]\n\n ::\n\n sage: R.<t> = QQ[]\n sage: s = SymmetricFunctions(R).s()\n sage: a = s([3])\n sage: f = t*s([2])\n sage: a(f)\n t^3*s[2, 2, 2] + t^3*s[4, 2] + t^3*s[6]\n sage: f(a)\n t*s[4, 2] + t*s[6]\n sage: s(0).plethysm(s[1])\n 0\n sage: s(1).plethysm(s[1])\n s[]\n sage: s(1).plethysm(s(0))\n s[]\n\n .. SEEALSO::\n\n :meth:`frobenius`\n "
if (not is_SymmetricFunction(x)):
raise TypeError('only know how to compute plethysms between symmetric functions')
parent = self.parent()
p = parent.realization_of().power()
R = parent.base_ring()
p_x = p(x)
if (self == parent.zero()):
return self
if ((include is not None) and (exclude is not None)):
raise RuntimeError('include and exclude cannot both be specified')
try:
degree_one = [R(g) for g in R.variable_names_recursive()]
except AttributeError:
try:
degree_one = R.gens()
except NotImplementedError:
degree_one = []
if include:
degree_one = [R(g) for g in include]
if exclude:
degree_one = [g for g in degree_one if (g not in exclude)]
scale_part = (lambda n: (lambda m: m.__class__(m.parent(), [(i * n) for i in m])))
raise_c = (lambda n: (lambda c: c.subs(**dict(((str(g), (g ** n)) for g in degree_one if (g != 1))))))
pn_pleth = (lambda f, n: f.map_support(scale_part(n)))
f = (lambda part: prod((pn_pleth(p_x.map_coefficients(raise_c(i)), i) for i in part)))
return parent(p._apply_module_morphism(p(self), f))
|
def plethysm(self, x, include=None, exclude=None):
"\n Return the outer plethysm of ``self`` with ``x``. This is\n implemented only over base rings which are `\\QQ`-algebras.\n (To compute outer plethysms over general binomial rings, change\n bases to the fraction field.)\n\n By default, the degree one elements are taken to be the\n generators for the ``self``'s base ring. This setting can be\n modified by specifying the ``include`` and ``exclude`` keywords.\n\n INPUT:\n\n - ``x`` -- a symmetric function over the same base ring as\n ``self``\n\n - ``include`` -- a list of variables to be treated as\n degree one elements instead of the default degree one elements\n\n - ``exclude`` -- a list of variables to be excluded\n from the default degree one elements\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: s = Sym.s()\n sage: h = Sym.h()\n sage: s ( h([3])( h([2]) ) )\n s[2, 2, 2] + s[4, 2] + s[6]\n sage: p = Sym.p()\n sage: p([3])( s([2,1]) )\n 1/3*p[3, 3, 3] - 1/3*p[9]\n sage: e = Sym.e()\n sage: e([3])( e([2]) )\n e[3, 3] + e[4, 1, 1] - 2*e[4, 2] - e[5, 1] + e[6]\n\n ::\n\n sage: R.<t> = QQ[]\n sage: s = SymmetricFunctions(R).s()\n sage: a = s([3])\n sage: f = t*s([2])\n sage: a(f)\n t^3*s[2, 2, 2] + t^3*s[4, 2] + t^3*s[6]\n sage: f(a)\n t*s[4, 2] + t*s[6]\n sage: s(0).plethysm(s[1])\n 0\n sage: s(1).plethysm(s[1])\n s[]\n sage: s(1).plethysm(s(0))\n s[]\n\n .. SEEALSO::\n\n :meth:`frobenius`\n "
if (not is_SymmetricFunction(x)):
raise TypeError('only know how to compute plethysms between symmetric functions')
parent = self.parent()
p = parent.realization_of().power()
R = parent.base_ring()
p_x = p(x)
if (self == parent.zero()):
return self
if ((include is not None) and (exclude is not None)):
raise RuntimeError('include and exclude cannot both be specified')
try:
degree_one = [R(g) for g in R.variable_names_recursive()]
except AttributeError:
try:
degree_one = R.gens()
except NotImplementedError:
degree_one = []
if include:
degree_one = [R(g) for g in include]
if exclude:
degree_one = [g for g in degree_one if (g not in exclude)]
scale_part = (lambda n: (lambda m: m.__class__(m.parent(), [(i * n) for i in m])))
raise_c = (lambda n: (lambda c: c.subs(**dict(((str(g), (g ** n)) for g in degree_one if (g != 1))))))
pn_pleth = (lambda f, n: f.map_support(scale_part(n)))
f = (lambda part: prod((pn_pleth(p_x.map_coefficients(raise_c(i)), i) for i in part)))
return parent(p._apply_module_morphism(p(self), f))<|docstring|>Return the outer plethysm of ``self`` with ``x``. This is
implemented only over base rings which are `\QQ`-algebras.
(To compute outer plethysms over general binomial rings, change
bases to the fraction field.)
By default, the degree one elements are taken to be the
generators for the ``self``'s base ring. This setting can be
modified by specifying the ``include`` and ``exclude`` keywords.
INPUT:
- ``x`` -- a symmetric function over the same base ring as
``self``
- ``include`` -- a list of variables to be treated as
degree one elements instead of the default degree one elements
- ``exclude`` -- a list of variables to be excluded
from the default degree one elements
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.s()
sage: h = Sym.h()
sage: s ( h([3])( h([2]) ) )
s[2, 2, 2] + s[4, 2] + s[6]
sage: p = Sym.p()
sage: p([3])( s([2,1]) )
1/3*p[3, 3, 3] - 1/3*p[9]
sage: e = Sym.e()
sage: e([3])( e([2]) )
e[3, 3] + e[4, 1, 1] - 2*e[4, 2] - e[5, 1] + e[6]
::
sage: R.<t> = QQ[]
sage: s = SymmetricFunctions(R).s()
sage: a = s([3])
sage: f = t*s([2])
sage: a(f)
t^3*s[2, 2, 2] + t^3*s[4, 2] + t^3*s[6]
sage: f(a)
t*s[4, 2] + t*s[6]
sage: s(0).plethysm(s[1])
0
sage: s(1).plethysm(s[1])
s[]
sage: s(1).plethysm(s(0))
s[]
.. SEEALSO::
:meth:`frobenius`<|endoftext|>
|
83f50dd535aa518bb577e93e8ba9c1feb124cbd6df672b01055519a39409285e
|
def inner_plethysm(self, x):
"\n Return the inner plethysm of ``self`` with ``x``.\n\n Whenever `R` is a `\\QQ`-algebra, and `f` and `g` are two\n symmetric functions over `R` such that the constant term of `f`\n is zero, the inner plethysm of `f` with `g` is a symmetric\n function over `R`, and the degree of this symmetric function is\n the same as the degree of `g`. We will denote the inner plethysm\n of `f` with `g` by `f \\{ g \\}` (in contrast to the notation of\n outer plethysm which is generally denoted `f [ g ]`); in Sage\n syntax, it is ``f.inner_plethysm(g)``.\n\n First we describe the axiomatic definition of the operation; see\n below for a representation-theoretic interpretation.\n In the following equations, we denote the outer product\n (i.e., the standard product on the ring of symmetric functions,\n :meth:`~sage.categories.algebras_with_basis.AlgebrasWithBasis.ParentMethods.product`)\n by `\\cdot` and the Kronecker product (:meth:`itensor`) by `\\ast`).\n\n .. MATH::\n\n (f + g) \\{ h \\} = f \\{ h \\} + g \\{ h \\}\n\n (f \\cdot g) \\{ h \\} = (f \\{ h \\}) \\ast (g \\{ h \\})\n\n p_k \\{ f + g \\} = p_k \\{ f \\} + p_k \\{ g \\}\n\n where `p_k` is the `k`-th power-sum symmetric function for every\n `k > 0`.\n\n Let `\\sigma` be a permutation of cycle type `\\mu` and let `\\mu^k`\n be the cycle type of `\\sigma^k`. Then,\n\n .. MATH::\n\n p_k \\{ p_\\mu/z_\\mu \\} = \\sum_{\\nu : \\nu^k = \\mu } p_{\\nu}/z_{\\nu}\n\n Since `(p_\\mu/z_\\mu)_{\\mu}` is a basis for the symmetric\n functions, these four formulas define the symmetric function\n operation `f \\{ g \\}` for any symmetric functions `f` and `g`\n (where `f` has constant term `0`) by expanding `f` in the\n power sum basis and `g` in the dual basis `p_\\mu/z_\\mu`.\n\n .. SEEALSO:: :meth:`itensor`, :func:`~sage.combinat.partition.partition_power`,\n :meth:`plethysm`\n\n This operation admits a representation-theoretic interpretation\n in the case where `f` is a Schur function `s_\\lambda` and\n `g` is a homogeneous degree `n` symmetric function with\n nonnegative integral coefficients in the Schur basis.\n The symmetric function `f \\{ g \\}` is the Frobenius\n image of the `S_n`-representation constructed as follows.\n\n The assumptions on `g` imply that `g` is the Frobenius image of a\n representation `\\rho` of the symmetric group `S_n`:\n\n .. MATH::\n\n \\rho : S_n \\to GL_N.\n\n If the degree `N` of this representation is greater than or equal\n to the number of parts of `\\lambda`, then `f`, which denotes `s_\\lambda`,\n corresponds to the character of some irreducible `GL_N`-representation, say\n\n .. MATH::\n\n \\sigma : GL_N \\to GL_M.\n\n The composition `\\sigma \\circ \\rho : S_n \\to GL_M` is a representation\n of `S_n` whose Frobenius image is precisely `f \\{ g \\}`.\n\n If `N` is less than the number of parts of `\\lambda`,\n then `f \\{ g \\}` is `0` by definition.\n\n When `f` is a symmetric function with constant term `\\neq 0`, the\n inner plethysm `f \\{ g \\}` isn't well-defined in the ring of\n symmetric functions. Indeed, it is not clear how to define\n `1 \\{ g \\}`. The most sensible way to get around this probably is\n defining it as the infinite sum `h_0 + h_1 + h_2 + \\cdots` (where\n `h_i` means the `i`-th complete homogeneous symmetric function)\n in the completion of this ring with respect to its grading. This is\n how [SchaThi1994]_ defines `1 \\{ g \\}`. The present method,\n however, sets it to be the sum of `h_i` over all `i` for which the\n `i`-th homogeneous component of `g` is nonzero. This is rather a\n hack than a reasonable definition. Use with caution!\n\n .. NOTE::\n\n If a symmetric function `g` is written in the form\n `g = g_0 + g_1 + g_2 + \\cdots` with each `g_i` homogeneous\n of degree `i`, then\n `f \\{ g \\} = f \\{ g_0 \\} + f \\{ g_1 \\} + f \\{ g_2 \\} + \\cdots`\n for every `f` with constant term `0`. But in general, inner\n plethysm is not linear in the second variable.\n\n REFERENCES:\n\n .. [King] King, R. Branching rules for `GL_m \\supset \\Sigma_n`\n and the evaluation of inner plethysms.\n J. Math. Phys. 15, 258 (1974) :doi:`10.1063/1.1666632`\n\n .. [SchaThi1994] Thomas Scharf, Jean-Yves Thibon.\n *A Hopf-algebra approach to inner plethysm*.\n Advances in Mathematics 104 (1994), pp. 30-58.\n ftp://ftp.mathe2.uni-bayreuth.de/axel/papers/scharf:a_hopf_algebra_approach_to_inner_plethysm.ps.gz\n\n INPUT:\n\n - ``x`` -- element of the ring of symmetric functions over the same\n base ring as ``self``\n\n OUTPUT:\n\n - an element of symmetric functions in the parent of ``self``\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: s = Sym.schur()\n sage: p = Sym.power()\n sage: h = Sym.complete()\n sage: s([2,1]).inner_plethysm(s([1,1,1]))\n 0\n sage: s([2]).inner_plethysm(s([2,1]))\n s[2, 1] + s[3]\n sage: s([1,1]).inner_plethysm(s([2,1]))\n s[1, 1, 1]\n sage: s[2,1].inner_tensor(s[2,1])\n s[1, 1, 1] + s[2, 1] + s[3]\n\n ::\n\n sage: f = s([2,1]) + 2*s([3,1])\n sage: f.itensor(f)\n s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]\n sage: s( h([1,1]).inner_plethysm(f) )\n s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]\n\n ::\n\n sage: s([]).inner_plethysm(s([1,1]) + 2*s([2,1])+s([3]))\n s[2] + s[3]\n sage: [s([]).inner_plethysm(s(la)) for la in Partitions(4)]\n [s[4], s[4], s[4], s[4], s[4]]\n sage: s([3]).inner_plethysm(s([]))\n s[]\n sage: s[1,1,1,1].inner_plethysm(s[2,1])\n 0\n sage: s[1,1,1,1].inner_plethysm(2*s[2,1])\n s[3]\n\n ::\n\n sage: p[3].inner_plethysm(p[3])\n 0\n sage: p[3,3].inner_plethysm(p[3])\n 0\n sage: p[3].inner_plethysm(p[1,1,1])\n p[1, 1, 1] + 2*p[3]\n sage: p[4].inner_plethysm(p[1,1,1,1]/24)\n 1/24*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] + 1/8*p[2, 2] + 1/4*p[4]\n sage: p[3,3].inner_plethysm(p[1,1,1])\n 6*p[1, 1, 1] + 12*p[3]\n\n TESTS::\n\n sage: s(0).inner_plethysm(s(0))\n 0\n sage: s(1).inner_plethysm(s(0))\n 0\n sage: s(0).inner_plethysm(s(1))\n 0\n sage: s(1).inner_plethysm(s(1))\n s[]\n sage: s(2).inner_plethysm(s(1))\n 2*s[]\n sage: s(1).inner_plethysm(s(2))\n s[]\n "
parent = self.parent()
if (self == parent.zero()):
return self
p = parent.realization_of().power()
cache = {}
ip_pnu_g = parent._inner_plethysm_pnu_g
return sum(((c * ip_pnu_g(p(x), cache, nu)) for (nu, c) in p(self).monomial_coefficients().iteritems()))
|
Return the inner plethysm of ``self`` with ``x``.
Whenever `R` is a `\QQ`-algebra, and `f` and `g` are two
symmetric functions over `R` such that the constant term of `f`
is zero, the inner plethysm of `f` with `g` is a symmetric
function over `R`, and the degree of this symmetric function is
the same as the degree of `g`. We will denote the inner plethysm
of `f` with `g` by `f \{ g \}` (in contrast to the notation of
outer plethysm which is generally denoted `f [ g ]`); in Sage
syntax, it is ``f.inner_plethysm(g)``.
First we describe the axiomatic definition of the operation; see
below for a representation-theoretic interpretation.
In the following equations, we denote the outer product
(i.e., the standard product on the ring of symmetric functions,
:meth:`~sage.categories.algebras_with_basis.AlgebrasWithBasis.ParentMethods.product`)
by `\cdot` and the Kronecker product (:meth:`itensor`) by `\ast`).
.. MATH::
(f + g) \{ h \} = f \{ h \} + g \{ h \}
(f \cdot g) \{ h \} = (f \{ h \}) \ast (g \{ h \})
p_k \{ f + g \} = p_k \{ f \} + p_k \{ g \}
where `p_k` is the `k`-th power-sum symmetric function for every
`k > 0`.
Let `\sigma` be a permutation of cycle type `\mu` and let `\mu^k`
be the cycle type of `\sigma^k`. Then,
.. MATH::
p_k \{ p_\mu/z_\mu \} = \sum_{\nu : \nu^k = \mu } p_{\nu}/z_{\nu}
Since `(p_\mu/z_\mu)_{\mu}` is a basis for the symmetric
functions, these four formulas define the symmetric function
operation `f \{ g \}` for any symmetric functions `f` and `g`
(where `f` has constant term `0`) by expanding `f` in the
power sum basis and `g` in the dual basis `p_\mu/z_\mu`.
.. SEEALSO:: :meth:`itensor`, :func:`~sage.combinat.partition.partition_power`,
:meth:`plethysm`
This operation admits a representation-theoretic interpretation
in the case where `f` is a Schur function `s_\lambda` and
`g` is a homogeneous degree `n` symmetric function with
nonnegative integral coefficients in the Schur basis.
The symmetric function `f \{ g \}` is the Frobenius
image of the `S_n`-representation constructed as follows.
The assumptions on `g` imply that `g` is the Frobenius image of a
representation `\rho` of the symmetric group `S_n`:
.. MATH::
\rho : S_n \to GL_N.
If the degree `N` of this representation is greater than or equal
to the number of parts of `\lambda`, then `f`, which denotes `s_\lambda`,
corresponds to the character of some irreducible `GL_N`-representation, say
.. MATH::
\sigma : GL_N \to GL_M.
The composition `\sigma \circ \rho : S_n \to GL_M` is a representation
of `S_n` whose Frobenius image is precisely `f \{ g \}`.
If `N` is less than the number of parts of `\lambda`,
then `f \{ g \}` is `0` by definition.
When `f` is a symmetric function with constant term `\neq 0`, the
inner plethysm `f \{ g \}` isn't well-defined in the ring of
symmetric functions. Indeed, it is not clear how to define
`1 \{ g \}`. The most sensible way to get around this probably is
defining it as the infinite sum `h_0 + h_1 + h_2 + \cdots` (where
`h_i` means the `i`-th complete homogeneous symmetric function)
in the completion of this ring with respect to its grading. This is
how [SchaThi1994]_ defines `1 \{ g \}`. The present method,
however, sets it to be the sum of `h_i` over all `i` for which the
`i`-th homogeneous component of `g` is nonzero. This is rather a
hack than a reasonable definition. Use with caution!
.. NOTE::
If a symmetric function `g` is written in the form
`g = g_0 + g_1 + g_2 + \cdots` with each `g_i` homogeneous
of degree `i`, then
`f \{ g \} = f \{ g_0 \} + f \{ g_1 \} + f \{ g_2 \} + \cdots`
for every `f` with constant term `0`. But in general, inner
plethysm is not linear in the second variable.
REFERENCES:
.. [King] King, R. Branching rules for `GL_m \supset \Sigma_n`
and the evaluation of inner plethysms.
J. Math. Phys. 15, 258 (1974) :doi:`10.1063/1.1666632`
.. [SchaThi1994] Thomas Scharf, Jean-Yves Thibon.
*A Hopf-algebra approach to inner plethysm*.
Advances in Mathematics 104 (1994), pp. 30-58.
ftp://ftp.mathe2.uni-bayreuth.de/axel/papers/scharf:a_hopf_algebra_approach_to_inner_plethysm.ps.gz
INPUT:
- ``x`` -- element of the ring of symmetric functions over the same
base ring as ``self``
OUTPUT:
- an element of symmetric functions in the parent of ``self``
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.schur()
sage: p = Sym.power()
sage: h = Sym.complete()
sage: s([2,1]).inner_plethysm(s([1,1,1]))
0
sage: s([2]).inner_plethysm(s([2,1]))
s[2, 1] + s[3]
sage: s([1,1]).inner_plethysm(s([2,1]))
s[1, 1, 1]
sage: s[2,1].inner_tensor(s[2,1])
s[1, 1, 1] + s[2, 1] + s[3]
::
sage: f = s([2,1]) + 2*s([3,1])
sage: f.itensor(f)
s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]
sage: s( h([1,1]).inner_plethysm(f) )
s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]
::
sage: s([]).inner_plethysm(s([1,1]) + 2*s([2,1])+s([3]))
s[2] + s[3]
sage: [s([]).inner_plethysm(s(la)) for la in Partitions(4)]
[s[4], s[4], s[4], s[4], s[4]]
sage: s([3]).inner_plethysm(s([]))
s[]
sage: s[1,1,1,1].inner_plethysm(s[2,1])
0
sage: s[1,1,1,1].inner_plethysm(2*s[2,1])
s[3]
::
sage: p[3].inner_plethysm(p[3])
0
sage: p[3,3].inner_plethysm(p[3])
0
sage: p[3].inner_plethysm(p[1,1,1])
p[1, 1, 1] + 2*p[3]
sage: p[4].inner_plethysm(p[1,1,1,1]/24)
1/24*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] + 1/8*p[2, 2] + 1/4*p[4]
sage: p[3,3].inner_plethysm(p[1,1,1])
6*p[1, 1, 1] + 12*p[3]
TESTS::
sage: s(0).inner_plethysm(s(0))
0
sage: s(1).inner_plethysm(s(0))
0
sage: s(0).inner_plethysm(s(1))
0
sage: s(1).inner_plethysm(s(1))
s[]
sage: s(2).inner_plethysm(s(1))
2*s[]
sage: s(1).inner_plethysm(s(2))
s[]
|
src/sage/combinat/sf/sfa.py
|
inner_plethysm
|
bopopescu/sagesmc
| 5 |
python
|
def inner_plethysm(self, x):
"\n Return the inner plethysm of ``self`` with ``x``.\n\n Whenever `R` is a `\\QQ`-algebra, and `f` and `g` are two\n symmetric functions over `R` such that the constant term of `f`\n is zero, the inner plethysm of `f` with `g` is a symmetric\n function over `R`, and the degree of this symmetric function is\n the same as the degree of `g`. We will denote the inner plethysm\n of `f` with `g` by `f \\{ g \\}` (in contrast to the notation of\n outer plethysm which is generally denoted `f [ g ]`); in Sage\n syntax, it is ``f.inner_plethysm(g)``.\n\n First we describe the axiomatic definition of the operation; see\n below for a representation-theoretic interpretation.\n In the following equations, we denote the outer product\n (i.e., the standard product on the ring of symmetric functions,\n :meth:`~sage.categories.algebras_with_basis.AlgebrasWithBasis.ParentMethods.product`)\n by `\\cdot` and the Kronecker product (:meth:`itensor`) by `\\ast`).\n\n .. MATH::\n\n (f + g) \\{ h \\} = f \\{ h \\} + g \\{ h \\}\n\n (f \\cdot g) \\{ h \\} = (f \\{ h \\}) \\ast (g \\{ h \\})\n\n p_k \\{ f + g \\} = p_k \\{ f \\} + p_k \\{ g \\}\n\n where `p_k` is the `k`-th power-sum symmetric function for every\n `k > 0`.\n\n Let `\\sigma` be a permutation of cycle type `\\mu` and let `\\mu^k`\n be the cycle type of `\\sigma^k`. Then,\n\n .. MATH::\n\n p_k \\{ p_\\mu/z_\\mu \\} = \\sum_{\\nu : \\nu^k = \\mu } p_{\\nu}/z_{\\nu}\n\n Since `(p_\\mu/z_\\mu)_{\\mu}` is a basis for the symmetric\n functions, these four formulas define the symmetric function\n operation `f \\{ g \\}` for any symmetric functions `f` and `g`\n (where `f` has constant term `0`) by expanding `f` in the\n power sum basis and `g` in the dual basis `p_\\mu/z_\\mu`.\n\n .. SEEALSO:: :meth:`itensor`, :func:`~sage.combinat.partition.partition_power`,\n :meth:`plethysm`\n\n This operation admits a representation-theoretic interpretation\n in the case where `f` is a Schur function `s_\\lambda` and\n `g` is a homogeneous degree `n` symmetric function with\n nonnegative integral coefficients in the Schur basis.\n The symmetric function `f \\{ g \\}` is the Frobenius\n image of the `S_n`-representation constructed as follows.\n\n The assumptions on `g` imply that `g` is the Frobenius image of a\n representation `\\rho` of the symmetric group `S_n`:\n\n .. MATH::\n\n \\rho : S_n \\to GL_N.\n\n If the degree `N` of this representation is greater than or equal\n to the number of parts of `\\lambda`, then `f`, which denotes `s_\\lambda`,\n corresponds to the character of some irreducible `GL_N`-representation, say\n\n .. MATH::\n\n \\sigma : GL_N \\to GL_M.\n\n The composition `\\sigma \\circ \\rho : S_n \\to GL_M` is a representation\n of `S_n` whose Frobenius image is precisely `f \\{ g \\}`.\n\n If `N` is less than the number of parts of `\\lambda`,\n then `f \\{ g \\}` is `0` by definition.\n\n When `f` is a symmetric function with constant term `\\neq 0`, the\n inner plethysm `f \\{ g \\}` isn't well-defined in the ring of\n symmetric functions. Indeed, it is not clear how to define\n `1 \\{ g \\}`. The most sensible way to get around this probably is\n defining it as the infinite sum `h_0 + h_1 + h_2 + \\cdots` (where\n `h_i` means the `i`-th complete homogeneous symmetric function)\n in the completion of this ring with respect to its grading. This is\n how [SchaThi1994]_ defines `1 \\{ g \\}`. The present method,\n however, sets it to be the sum of `h_i` over all `i` for which the\n `i`-th homogeneous component of `g` is nonzero. This is rather a\n hack than a reasonable definition. Use with caution!\n\n .. NOTE::\n\n If a symmetric function `g` is written in the form\n `g = g_0 + g_1 + g_2 + \\cdots` with each `g_i` homogeneous\n of degree `i`, then\n `f \\{ g \\} = f \\{ g_0 \\} + f \\{ g_1 \\} + f \\{ g_2 \\} + \\cdots`\n for every `f` with constant term `0`. But in general, inner\n plethysm is not linear in the second variable.\n\n REFERENCES:\n\n .. [King] King, R. Branching rules for `GL_m \\supset \\Sigma_n`\n and the evaluation of inner plethysms.\n J. Math. Phys. 15, 258 (1974) :doi:`10.1063/1.1666632`\n\n .. [SchaThi1994] Thomas Scharf, Jean-Yves Thibon.\n *A Hopf-algebra approach to inner plethysm*.\n Advances in Mathematics 104 (1994), pp. 30-58.\n ftp://ftp.mathe2.uni-bayreuth.de/axel/papers/scharf:a_hopf_algebra_approach_to_inner_plethysm.ps.gz\n\n INPUT:\n\n - ``x`` -- element of the ring of symmetric functions over the same\n base ring as ``self``\n\n OUTPUT:\n\n - an element of symmetric functions in the parent of ``self``\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: s = Sym.schur()\n sage: p = Sym.power()\n sage: h = Sym.complete()\n sage: s([2,1]).inner_plethysm(s([1,1,1]))\n 0\n sage: s([2]).inner_plethysm(s([2,1]))\n s[2, 1] + s[3]\n sage: s([1,1]).inner_plethysm(s([2,1]))\n s[1, 1, 1]\n sage: s[2,1].inner_tensor(s[2,1])\n s[1, 1, 1] + s[2, 1] + s[3]\n\n ::\n\n sage: f = s([2,1]) + 2*s([3,1])\n sage: f.itensor(f)\n s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]\n sage: s( h([1,1]).inner_plethysm(f) )\n s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]\n\n ::\n\n sage: s([]).inner_plethysm(s([1,1]) + 2*s([2,1])+s([3]))\n s[2] + s[3]\n sage: [s([]).inner_plethysm(s(la)) for la in Partitions(4)]\n [s[4], s[4], s[4], s[4], s[4]]\n sage: s([3]).inner_plethysm(s([]))\n s[]\n sage: s[1,1,1,1].inner_plethysm(s[2,1])\n 0\n sage: s[1,1,1,1].inner_plethysm(2*s[2,1])\n s[3]\n\n ::\n\n sage: p[3].inner_plethysm(p[3])\n 0\n sage: p[3,3].inner_plethysm(p[3])\n 0\n sage: p[3].inner_plethysm(p[1,1,1])\n p[1, 1, 1] + 2*p[3]\n sage: p[4].inner_plethysm(p[1,1,1,1]/24)\n 1/24*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] + 1/8*p[2, 2] + 1/4*p[4]\n sage: p[3,3].inner_plethysm(p[1,1,1])\n 6*p[1, 1, 1] + 12*p[3]\n\n TESTS::\n\n sage: s(0).inner_plethysm(s(0))\n 0\n sage: s(1).inner_plethysm(s(0))\n 0\n sage: s(0).inner_plethysm(s(1))\n 0\n sage: s(1).inner_plethysm(s(1))\n s[]\n sage: s(2).inner_plethysm(s(1))\n 2*s[]\n sage: s(1).inner_plethysm(s(2))\n s[]\n "
parent = self.parent()
if (self == parent.zero()):
return self
p = parent.realization_of().power()
cache = {}
ip_pnu_g = parent._inner_plethysm_pnu_g
return sum(((c * ip_pnu_g(p(x), cache, nu)) for (nu, c) in p(self).monomial_coefficients().iteritems()))
|
def inner_plethysm(self, x):
"\n Return the inner plethysm of ``self`` with ``x``.\n\n Whenever `R` is a `\\QQ`-algebra, and `f` and `g` are two\n symmetric functions over `R` such that the constant term of `f`\n is zero, the inner plethysm of `f` with `g` is a symmetric\n function over `R`, and the degree of this symmetric function is\n the same as the degree of `g`. We will denote the inner plethysm\n of `f` with `g` by `f \\{ g \\}` (in contrast to the notation of\n outer plethysm which is generally denoted `f [ g ]`); in Sage\n syntax, it is ``f.inner_plethysm(g)``.\n\n First we describe the axiomatic definition of the operation; see\n below for a representation-theoretic interpretation.\n In the following equations, we denote the outer product\n (i.e., the standard product on the ring of symmetric functions,\n :meth:`~sage.categories.algebras_with_basis.AlgebrasWithBasis.ParentMethods.product`)\n by `\\cdot` and the Kronecker product (:meth:`itensor`) by `\\ast`).\n\n .. MATH::\n\n (f + g) \\{ h \\} = f \\{ h \\} + g \\{ h \\}\n\n (f \\cdot g) \\{ h \\} = (f \\{ h \\}) \\ast (g \\{ h \\})\n\n p_k \\{ f + g \\} = p_k \\{ f \\} + p_k \\{ g \\}\n\n where `p_k` is the `k`-th power-sum symmetric function for every\n `k > 0`.\n\n Let `\\sigma` be a permutation of cycle type `\\mu` and let `\\mu^k`\n be the cycle type of `\\sigma^k`. Then,\n\n .. MATH::\n\n p_k \\{ p_\\mu/z_\\mu \\} = \\sum_{\\nu : \\nu^k = \\mu } p_{\\nu}/z_{\\nu}\n\n Since `(p_\\mu/z_\\mu)_{\\mu}` is a basis for the symmetric\n functions, these four formulas define the symmetric function\n operation `f \\{ g \\}` for any symmetric functions `f` and `g`\n (where `f` has constant term `0`) by expanding `f` in the\n power sum basis and `g` in the dual basis `p_\\mu/z_\\mu`.\n\n .. SEEALSO:: :meth:`itensor`, :func:`~sage.combinat.partition.partition_power`,\n :meth:`plethysm`\n\n This operation admits a representation-theoretic interpretation\n in the case where `f` is a Schur function `s_\\lambda` and\n `g` is a homogeneous degree `n` symmetric function with\n nonnegative integral coefficients in the Schur basis.\n The symmetric function `f \\{ g \\}` is the Frobenius\n image of the `S_n`-representation constructed as follows.\n\n The assumptions on `g` imply that `g` is the Frobenius image of a\n representation `\\rho` of the symmetric group `S_n`:\n\n .. MATH::\n\n \\rho : S_n \\to GL_N.\n\n If the degree `N` of this representation is greater than or equal\n to the number of parts of `\\lambda`, then `f`, which denotes `s_\\lambda`,\n corresponds to the character of some irreducible `GL_N`-representation, say\n\n .. MATH::\n\n \\sigma : GL_N \\to GL_M.\n\n The composition `\\sigma \\circ \\rho : S_n \\to GL_M` is a representation\n of `S_n` whose Frobenius image is precisely `f \\{ g \\}`.\n\n If `N` is less than the number of parts of `\\lambda`,\n then `f \\{ g \\}` is `0` by definition.\n\n When `f` is a symmetric function with constant term `\\neq 0`, the\n inner plethysm `f \\{ g \\}` isn't well-defined in the ring of\n symmetric functions. Indeed, it is not clear how to define\n `1 \\{ g \\}`. The most sensible way to get around this probably is\n defining it as the infinite sum `h_0 + h_1 + h_2 + \\cdots` (where\n `h_i` means the `i`-th complete homogeneous symmetric function)\n in the completion of this ring with respect to its grading. This is\n how [SchaThi1994]_ defines `1 \\{ g \\}`. The present method,\n however, sets it to be the sum of `h_i` over all `i` for which the\n `i`-th homogeneous component of `g` is nonzero. This is rather a\n hack than a reasonable definition. Use with caution!\n\n .. NOTE::\n\n If a symmetric function `g` is written in the form\n `g = g_0 + g_1 + g_2 + \\cdots` with each `g_i` homogeneous\n of degree `i`, then\n `f \\{ g \\} = f \\{ g_0 \\} + f \\{ g_1 \\} + f \\{ g_2 \\} + \\cdots`\n for every `f` with constant term `0`. But in general, inner\n plethysm is not linear in the second variable.\n\n REFERENCES:\n\n .. [King] King, R. Branching rules for `GL_m \\supset \\Sigma_n`\n and the evaluation of inner plethysms.\n J. Math. Phys. 15, 258 (1974) :doi:`10.1063/1.1666632`\n\n .. [SchaThi1994] Thomas Scharf, Jean-Yves Thibon.\n *A Hopf-algebra approach to inner plethysm*.\n Advances in Mathematics 104 (1994), pp. 30-58.\n ftp://ftp.mathe2.uni-bayreuth.de/axel/papers/scharf:a_hopf_algebra_approach_to_inner_plethysm.ps.gz\n\n INPUT:\n\n - ``x`` -- element of the ring of symmetric functions over the same\n base ring as ``self``\n\n OUTPUT:\n\n - an element of symmetric functions in the parent of ``self``\n\n EXAMPLES::\n\n sage: Sym = SymmetricFunctions(QQ)\n sage: s = Sym.schur()\n sage: p = Sym.power()\n sage: h = Sym.complete()\n sage: s([2,1]).inner_plethysm(s([1,1,1]))\n 0\n sage: s([2]).inner_plethysm(s([2,1]))\n s[2, 1] + s[3]\n sage: s([1,1]).inner_plethysm(s([2,1]))\n s[1, 1, 1]\n sage: s[2,1].inner_tensor(s[2,1])\n s[1, 1, 1] + s[2, 1] + s[3]\n\n ::\n\n sage: f = s([2,1]) + 2*s([3,1])\n sage: f.itensor(f)\n s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]\n sage: s( h([1,1]).inner_plethysm(f) )\n s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]\n\n ::\n\n sage: s([]).inner_plethysm(s([1,1]) + 2*s([2,1])+s([3]))\n s[2] + s[3]\n sage: [s([]).inner_plethysm(s(la)) for la in Partitions(4)]\n [s[4], s[4], s[4], s[4], s[4]]\n sage: s([3]).inner_plethysm(s([]))\n s[]\n sage: s[1,1,1,1].inner_plethysm(s[2,1])\n 0\n sage: s[1,1,1,1].inner_plethysm(2*s[2,1])\n s[3]\n\n ::\n\n sage: p[3].inner_plethysm(p[3])\n 0\n sage: p[3,3].inner_plethysm(p[3])\n 0\n sage: p[3].inner_plethysm(p[1,1,1])\n p[1, 1, 1] + 2*p[3]\n sage: p[4].inner_plethysm(p[1,1,1,1]/24)\n 1/24*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] + 1/8*p[2, 2] + 1/4*p[4]\n sage: p[3,3].inner_plethysm(p[1,1,1])\n 6*p[1, 1, 1] + 12*p[3]\n\n TESTS::\n\n sage: s(0).inner_plethysm(s(0))\n 0\n sage: s(1).inner_plethysm(s(0))\n 0\n sage: s(0).inner_plethysm(s(1))\n 0\n sage: s(1).inner_plethysm(s(1))\n s[]\n sage: s(2).inner_plethysm(s(1))\n 2*s[]\n sage: s(1).inner_plethysm(s(2))\n s[]\n "
parent = self.parent()
if (self == parent.zero()):
return self
p = parent.realization_of().power()
cache = {}
ip_pnu_g = parent._inner_plethysm_pnu_g
return sum(((c * ip_pnu_g(p(x), cache, nu)) for (nu, c) in p(self).monomial_coefficients().iteritems()))<|docstring|>Return the inner plethysm of ``self`` with ``x``.
Whenever `R` is a `\QQ`-algebra, and `f` and `g` are two
symmetric functions over `R` such that the constant term of `f`
is zero, the inner plethysm of `f` with `g` is a symmetric
function over `R`, and the degree of this symmetric function is
the same as the degree of `g`. We will denote the inner plethysm
of `f` with `g` by `f \{ g \}` (in contrast to the notation of
outer plethysm which is generally denoted `f [ g ]`); in Sage
syntax, it is ``f.inner_plethysm(g)``.
First we describe the axiomatic definition of the operation; see
below for a representation-theoretic interpretation.
In the following equations, we denote the outer product
(i.e., the standard product on the ring of symmetric functions,
:meth:`~sage.categories.algebras_with_basis.AlgebrasWithBasis.ParentMethods.product`)
by `\cdot` and the Kronecker product (:meth:`itensor`) by `\ast`).
.. MATH::
(f + g) \{ h \} = f \{ h \} + g \{ h \}
(f \cdot g) \{ h \} = (f \{ h \}) \ast (g \{ h \})
p_k \{ f + g \} = p_k \{ f \} + p_k \{ g \}
where `p_k` is the `k`-th power-sum symmetric function for every
`k > 0`.
Let `\sigma` be a permutation of cycle type `\mu` and let `\mu^k`
be the cycle type of `\sigma^k`. Then,
.. MATH::
p_k \{ p_\mu/z_\mu \} = \sum_{\nu : \nu^k = \mu } p_{\nu}/z_{\nu}
Since `(p_\mu/z_\mu)_{\mu}` is a basis for the symmetric
functions, these four formulas define the symmetric function
operation `f \{ g \}` for any symmetric functions `f` and `g`
(where `f` has constant term `0`) by expanding `f` in the
power sum basis and `g` in the dual basis `p_\mu/z_\mu`.
.. SEEALSO:: :meth:`itensor`, :func:`~sage.combinat.partition.partition_power`,
:meth:`plethysm`
This operation admits a representation-theoretic interpretation
in the case where `f` is a Schur function `s_\lambda` and
`g` is a homogeneous degree `n` symmetric function with
nonnegative integral coefficients in the Schur basis.
The symmetric function `f \{ g \}` is the Frobenius
image of the `S_n`-representation constructed as follows.
The assumptions on `g` imply that `g` is the Frobenius image of a
representation `\rho` of the symmetric group `S_n`:
.. MATH::
\rho : S_n \to GL_N.
If the degree `N` of this representation is greater than or equal
to the number of parts of `\lambda`, then `f`, which denotes `s_\lambda`,
corresponds to the character of some irreducible `GL_N`-representation, say
.. MATH::
\sigma : GL_N \to GL_M.
The composition `\sigma \circ \rho : S_n \to GL_M` is a representation
of `S_n` whose Frobenius image is precisely `f \{ g \}`.
If `N` is less than the number of parts of `\lambda`,
then `f \{ g \}` is `0` by definition.
When `f` is a symmetric function with constant term `\neq 0`, the
inner plethysm `f \{ g \}` isn't well-defined in the ring of
symmetric functions. Indeed, it is not clear how to define
`1 \{ g \}`. The most sensible way to get around this probably is
defining it as the infinite sum `h_0 + h_1 + h_2 + \cdots` (where
`h_i` means the `i`-th complete homogeneous symmetric function)
in the completion of this ring with respect to its grading. This is
how [SchaThi1994]_ defines `1 \{ g \}`. The present method,
however, sets it to be the sum of `h_i` over all `i` for which the
`i`-th homogeneous component of `g` is nonzero. This is rather a
hack than a reasonable definition. Use with caution!
.. NOTE::
If a symmetric function `g` is written in the form
`g = g_0 + g_1 + g_2 + \cdots` with each `g_i` homogeneous
of degree `i`, then
`f \{ g \} = f \{ g_0 \} + f \{ g_1 \} + f \{ g_2 \} + \cdots`
for every `f` with constant term `0`. But in general, inner
plethysm is not linear in the second variable.
REFERENCES:
.. [King] King, R. Branching rules for `GL_m \supset \Sigma_n`
and the evaluation of inner plethysms.
J. Math. Phys. 15, 258 (1974) :doi:`10.1063/1.1666632`
.. [SchaThi1994] Thomas Scharf, Jean-Yves Thibon.
*A Hopf-algebra approach to inner plethysm*.
Advances in Mathematics 104 (1994), pp. 30-58.
ftp://ftp.mathe2.uni-bayreuth.de/axel/papers/scharf:a_hopf_algebra_approach_to_inner_plethysm.ps.gz
INPUT:
- ``x`` -- element of the ring of symmetric functions over the same
base ring as ``self``
OUTPUT:
- an element of symmetric functions in the parent of ``self``
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.schur()
sage: p = Sym.power()
sage: h = Sym.complete()
sage: s([2,1]).inner_plethysm(s([1,1,1]))
0
sage: s([2]).inner_plethysm(s([2,1]))
s[2, 1] + s[3]
sage: s([1,1]).inner_plethysm(s([2,1]))
s[1, 1, 1]
sage: s[2,1].inner_tensor(s[2,1])
s[1, 1, 1] + s[2, 1] + s[3]
::
sage: f = s([2,1]) + 2*s([3,1])
sage: f.itensor(f)
s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]
sage: s( h([1,1]).inner_plethysm(f) )
s[1, 1, 1] + s[2, 1] + 4*s[2, 1, 1] + 4*s[2, 2] + s[3] + 4*s[3, 1] + 4*s[4]
::
sage: s([]).inner_plethysm(s([1,1]) + 2*s([2,1])+s([3]))
s[2] + s[3]
sage: [s([]).inner_plethysm(s(la)) for la in Partitions(4)]
[s[4], s[4], s[4], s[4], s[4]]
sage: s([3]).inner_plethysm(s([]))
s[]
sage: s[1,1,1,1].inner_plethysm(s[2,1])
0
sage: s[1,1,1,1].inner_plethysm(2*s[2,1])
s[3]
::
sage: p[3].inner_plethysm(p[3])
0
sage: p[3,3].inner_plethysm(p[3])
0
sage: p[3].inner_plethysm(p[1,1,1])
p[1, 1, 1] + 2*p[3]
sage: p[4].inner_plethysm(p[1,1,1,1]/24)
1/24*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] + 1/8*p[2, 2] + 1/4*p[4]
sage: p[3,3].inner_plethysm(p[1,1,1])
6*p[1, 1, 1] + 12*p[3]
TESTS::
sage: s(0).inner_plethysm(s(0))
0
sage: s(1).inner_plethysm(s(0))
0
sage: s(0).inner_plethysm(s(1))
0
sage: s(1).inner_plethysm(s(1))
s[]
sage: s(2).inner_plethysm(s(1))
2*s[]
sage: s(1).inner_plethysm(s(2))
s[]<|endoftext|>
|
5775d6225d67afcb048e2504600f8a279f572984ced27367e96a4f3a7f88177a
|
def omega(self):
'\n Return the image of ``self`` under the omega automorphism.\n\n The omega automorphism is defined to be the unique algebra\n endomorphism `\\omega` of the ring of symmetric functions that\n satisfies `\\omega(e_k) = h_k` for all positive integers `k`\n (where `e_k` stands for the `k`-th elementary symmetric\n function, and `h_k` stands for the `k`-th complete homogeneous\n symmetric function). It furthermore is a Hopf algebra\n endomorphism, and sends the power-sum symmetric function `p_k`\n to `(-1)^{k-1} p_k` for every positive integer `k`.\n\n The default implementation converts to the Schurs, then\n performs the automorphism and changes back.\n\n EXAMPLES::\n\n sage: J = SymmetricFunctions(QQ).jack(t=1).P()\n sage: a = J([2,1]) + J([1,1,1])\n sage: a.omega()\n JackP[2, 1] + JackP[3]\n sage: J(0).omega()\n 0\n sage: J(1).omega()\n JackP[]\n\n The forgotten symmetric functions are the images of the monomial\n symmetric functions under omega::\n\n sage: Sym = SymmetricFunctions(ZZ)\n sage: m = Sym.m()\n sage: f = Sym.f()\n sage: all( f(lam) == m(lam).omega() for lam in Partitions(3) )\n True\n sage: all( m(lam) == f(lam).omega() for lam in Partitions(3) )\n True\n '
parent = self.parent()
s = parent.realization_of().schur()
return parent(s(self).omega())
|
Return the image of ``self`` under the omega automorphism.
The omega automorphism is defined to be the unique algebra
endomorphism `\omega` of the ring of symmetric functions that
satisfies `\omega(e_k) = h_k` for all positive integers `k`
(where `e_k` stands for the `k`-th elementary symmetric
function, and `h_k` stands for the `k`-th complete homogeneous
symmetric function). It furthermore is a Hopf algebra
endomorphism, and sends the power-sum symmetric function `p_k`
to `(-1)^{k-1} p_k` for every positive integer `k`.
The default implementation converts to the Schurs, then
performs the automorphism and changes back.
EXAMPLES::
sage: J = SymmetricFunctions(QQ).jack(t=1).P()
sage: a = J([2,1]) + J([1,1,1])
sage: a.omega()
JackP[2, 1] + JackP[3]
sage: J(0).omega()
0
sage: J(1).omega()
JackP[]
The forgotten symmetric functions are the images of the monomial
symmetric functions under omega::
sage: Sym = SymmetricFunctions(ZZ)
sage: m = Sym.m()
sage: f = Sym.f()
sage: all( f(lam) == m(lam).omega() for lam in Partitions(3) )
True
sage: all( m(lam) == f(lam).omega() for lam in Partitions(3) )
True
|
src/sage/combinat/sf/sfa.py
|
omega
|
bopopescu/sagesmc
| 5 |
python
|
def omega(self):
'\n Return the image of ``self`` under the omega automorphism.\n\n The omega automorphism is defined to be the unique algebra\n endomorphism `\\omega` of the ring of symmetric functions that\n satisfies `\\omega(e_k) = h_k` for all positive integers `k`\n (where `e_k` stands for the `k`-th elementary symmetric\n function, and `h_k` stands for the `k`-th complete homogeneous\n symmetric function). It furthermore is a Hopf algebra\n endomorphism, and sends the power-sum symmetric function `p_k`\n to `(-1)^{k-1} p_k` for every positive integer `k`.\n\n The default implementation converts to the Schurs, then\n performs the automorphism and changes back.\n\n EXAMPLES::\n\n sage: J = SymmetricFunctions(QQ).jack(t=1).P()\n sage: a = J([2,1]) + J([1,1,1])\n sage: a.omega()\n JackP[2, 1] + JackP[3]\n sage: J(0).omega()\n 0\n sage: J(1).omega()\n JackP[]\n\n The forgotten symmetric functions are the images of the monomial\n symmetric functions under omega::\n\n sage: Sym = SymmetricFunctions(ZZ)\n sage: m = Sym.m()\n sage: f = Sym.f()\n sage: all( f(lam) == m(lam).omega() for lam in Partitions(3) )\n True\n sage: all( m(lam) == f(lam).omega() for lam in Partitions(3) )\n True\n '
parent = self.parent()
s = parent.realization_of().schur()
return parent(s(self).omega())
|
def omega(self):
'\n Return the image of ``self`` under the omega automorphism.\n\n The omega automorphism is defined to be the unique algebra\n endomorphism `\\omega` of the ring of symmetric functions that\n satisfies `\\omega(e_k) = h_k` for all positive integers `k`\n (where `e_k` stands for the `k`-th elementary symmetric\n function, and `h_k` stands for the `k`-th complete homogeneous\n symmetric function). It furthermore is a Hopf algebra\n endomorphism, and sends the power-sum symmetric function `p_k`\n to `(-1)^{k-1} p_k` for every positive integer `k`.\n\n The default implementation converts to the Schurs, then\n performs the automorphism and changes back.\n\n EXAMPLES::\n\n sage: J = SymmetricFunctions(QQ).jack(t=1).P()\n sage: a = J([2,1]) + J([1,1,1])\n sage: a.omega()\n JackP[2, 1] + JackP[3]\n sage: J(0).omega()\n 0\n sage: J(1).omega()\n JackP[]\n\n The forgotten symmetric functions are the images of the monomial\n symmetric functions under omega::\n\n sage: Sym = SymmetricFunctions(ZZ)\n sage: m = Sym.m()\n sage: f = Sym.f()\n sage: all( f(lam) == m(lam).omega() for lam in Partitions(3) )\n True\n sage: all( m(lam) == f(lam).omega() for lam in Partitions(3) )\n True\n '
parent = self.parent()
s = parent.realization_of().schur()
return parent(s(self).omega())<|docstring|>Return the image of ``self`` under the omega automorphism.
The omega automorphism is defined to be the unique algebra
endomorphism `\omega` of the ring of symmetric functions that
satisfies `\omega(e_k) = h_k` for all positive integers `k`
(where `e_k` stands for the `k`-th elementary symmetric
function, and `h_k` stands for the `k`-th complete homogeneous
symmetric function). It furthermore is a Hopf algebra
endomorphism, and sends the power-sum symmetric function `p_k`
to `(-1)^{k-1} p_k` for every positive integer `k`.
The default implementation converts to the Schurs, then
performs the automorphism and changes back.
EXAMPLES::
sage: J = SymmetricFunctions(QQ).jack(t=1).P()
sage: a = J([2,1]) + J([1,1,1])
sage: a.omega()
JackP[2, 1] + JackP[3]
sage: J(0).omega()
0
sage: J(1).omega()
JackP[]
The forgotten symmetric functions are the images of the monomial
symmetric functions under omega::
sage: Sym = SymmetricFunctions(ZZ)
sage: m = Sym.m()
sage: f = Sym.f()
sage: all( f(lam) == m(lam).omega() for lam in Partitions(3) )
True
sage: all( m(lam) == f(lam).omega() for lam in Partitions(3) )
True<|endoftext|>
|
a814f2aeb1711d7a653287bdcf5836ce467807e11256e2d079b9ceb730dbf1da
|
def theta(self, a):
'\n Return the image of ``self`` under the theta endomorphism which sends\n `p_k` to `a \\cdot p_k` for every positive integer `k`.\n\n In general, this is well-defined outside of the powersum basis only\n if the base ring is a `\\QQ`-algebra.\n\n INPUT:\n\n - ``a`` -- an element of the base ring\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s([2,1]).theta(2)\n 2*s[1, 1, 1] + 6*s[2, 1] + 2*s[3]\n sage: p = SymmetricFunctions(QQ).p()\n sage: p([2]).theta(2)\n 2*p[2]\n sage: p(0).theta(2)\n 0\n sage: p(1).theta(2)\n p[]\n '
p = self.parent().realization_of().power()
p_self = p(self)
res = p_self.map_item((lambda m, c: (m, (c * (a ** len(m))))))
return self.parent()(res)
|
Return the image of ``self`` under the theta endomorphism which sends
`p_k` to `a \cdot p_k` for every positive integer `k`.
In general, this is well-defined outside of the powersum basis only
if the base ring is a `\QQ`-algebra.
INPUT:
- ``a`` -- an element of the base ring
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: s([2,1]).theta(2)
2*s[1, 1, 1] + 6*s[2, 1] + 2*s[3]
sage: p = SymmetricFunctions(QQ).p()
sage: p([2]).theta(2)
2*p[2]
sage: p(0).theta(2)
0
sage: p(1).theta(2)
p[]
|
src/sage/combinat/sf/sfa.py
|
theta
|
bopopescu/sagesmc
| 5 |
python
|
def theta(self, a):
'\n Return the image of ``self`` under the theta endomorphism which sends\n `p_k` to `a \\cdot p_k` for every positive integer `k`.\n\n In general, this is well-defined outside of the powersum basis only\n if the base ring is a `\\QQ`-algebra.\n\n INPUT:\n\n - ``a`` -- an element of the base ring\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s([2,1]).theta(2)\n 2*s[1, 1, 1] + 6*s[2, 1] + 2*s[3]\n sage: p = SymmetricFunctions(QQ).p()\n sage: p([2]).theta(2)\n 2*p[2]\n sage: p(0).theta(2)\n 0\n sage: p(1).theta(2)\n p[]\n '
p = self.parent().realization_of().power()
p_self = p(self)
res = p_self.map_item((lambda m, c: (m, (c * (a ** len(m))))))
return self.parent()(res)
|
def theta(self, a):
'\n Return the image of ``self`` under the theta endomorphism which sends\n `p_k` to `a \\cdot p_k` for every positive integer `k`.\n\n In general, this is well-defined outside of the powersum basis only\n if the base ring is a `\\QQ`-algebra.\n\n INPUT:\n\n - ``a`` -- an element of the base ring\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: s([2,1]).theta(2)\n 2*s[1, 1, 1] + 6*s[2, 1] + 2*s[3]\n sage: p = SymmetricFunctions(QQ).p()\n sage: p([2]).theta(2)\n 2*p[2]\n sage: p(0).theta(2)\n 0\n sage: p(1).theta(2)\n p[]\n '
p = self.parent().realization_of().power()
p_self = p(self)
res = p_self.map_item((lambda m, c: (m, (c * (a ** len(m))))))
return self.parent()(res)<|docstring|>Return the image of ``self`` under the theta endomorphism which sends
`p_k` to `a \cdot p_k` for every positive integer `k`.
In general, this is well-defined outside of the powersum basis only
if the base ring is a `\QQ`-algebra.
INPUT:
- ``a`` -- an element of the base ring
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: s([2,1]).theta(2)
2*s[1, 1, 1] + 6*s[2, 1] + 2*s[3]
sage: p = SymmetricFunctions(QQ).p()
sage: p([2]).theta(2)
2*p[2]
sage: p(0).theta(2)
0
sage: p(1).theta(2)
p[]<|endoftext|>
|
e6bf47771e20526b4ef93ce2c061ae01b7eb47bac00b7f68168fd89d990a06ea
|
def theta_qt(self, q=None, t=None):
"\n Return the image of ``self`` under the `q,t`-deformed theta\n endomorphism which sends `p_k` to `\\frac{1-q^k}{1-t^k} \\cdot p_k`\n for all positive integers `k`.\n\n In general, this is well-defined outside of the powersum basis only\n if the base ring is a `\\QQ`-algebra.\n\n INPUT:\n\n - ``q``, ``t`` -- parameters (default: ``None``, in which case 'q'\n and 't' are used)\n\n EXAMPLES::\n\n sage: QQqt = QQ['q,t'].fraction_field()\n sage: q,t = QQqt.gens()\n sage: p = SymmetricFunctions(QQqt).p()\n sage: p([2]).theta_qt(q,t)\n ((-q^2+1)/(-t^2+1))*p[2]\n sage: p([2,1]).theta_qt(q,t)\n ((q^3-q^2-q+1)/(t^3-t^2-t+1))*p[2, 1]\n sage: p(0).theta_qt(q=1,t=3)\n 0\n sage: p([2,1]).theta_qt(q=2,t=3)\n 3/16*p[2, 1]\n sage: s = p.realization_of().schur()\n sage: s([3]).theta_qt(q=0)*(1-t)*(1-t^2)*(1-t^3)\n t^3*s[1, 1, 1] + (t^2+t)*s[2, 1] + s[3]\n sage: p(1).theta_qt()\n p[]\n "
parent = self.parent()
BR = parent.base_ring()
p = parent.realization_of().power()
p_self = p(self)
if (t is None):
if hasattr(parent, 't'):
t = parent.t
else:
t = BR(QQ['t'].gen())
if (q is None):
if hasattr(parent, 'q'):
q = parent.q
else:
q = BR(QQ['q'].gen())
res = p_self.map_item((lambda m, c: (m, BR((prod([((1 - (q ** k)) / (1 - (t ** k))) for k in m]) * c)))))
return parent(res)
|
Return the image of ``self`` under the `q,t`-deformed theta
endomorphism which sends `p_k` to `\frac{1-q^k}{1-t^k} \cdot p_k`
for all positive integers `k`.
In general, this is well-defined outside of the powersum basis only
if the base ring is a `\QQ`-algebra.
INPUT:
- ``q``, ``t`` -- parameters (default: ``None``, in which case 'q'
and 't' are used)
EXAMPLES::
sage: QQqt = QQ['q,t'].fraction_field()
sage: q,t = QQqt.gens()
sage: p = SymmetricFunctions(QQqt).p()
sage: p([2]).theta_qt(q,t)
((-q^2+1)/(-t^2+1))*p[2]
sage: p([2,1]).theta_qt(q,t)
((q^3-q^2-q+1)/(t^3-t^2-t+1))*p[2, 1]
sage: p(0).theta_qt(q=1,t=3)
0
sage: p([2,1]).theta_qt(q=2,t=3)
3/16*p[2, 1]
sage: s = p.realization_of().schur()
sage: s([3]).theta_qt(q=0)*(1-t)*(1-t^2)*(1-t^3)
t^3*s[1, 1, 1] + (t^2+t)*s[2, 1] + s[3]
sage: p(1).theta_qt()
p[]
|
src/sage/combinat/sf/sfa.py
|
theta_qt
|
bopopescu/sagesmc
| 5 |
python
|
def theta_qt(self, q=None, t=None):
"\n Return the image of ``self`` under the `q,t`-deformed theta\n endomorphism which sends `p_k` to `\\frac{1-q^k}{1-t^k} \\cdot p_k`\n for all positive integers `k`.\n\n In general, this is well-defined outside of the powersum basis only\n if the base ring is a `\\QQ`-algebra.\n\n INPUT:\n\n - ``q``, ``t`` -- parameters (default: ``None``, in which case 'q'\n and 't' are used)\n\n EXAMPLES::\n\n sage: QQqt = QQ['q,t'].fraction_field()\n sage: q,t = QQqt.gens()\n sage: p = SymmetricFunctions(QQqt).p()\n sage: p([2]).theta_qt(q,t)\n ((-q^2+1)/(-t^2+1))*p[2]\n sage: p([2,1]).theta_qt(q,t)\n ((q^3-q^2-q+1)/(t^3-t^2-t+1))*p[2, 1]\n sage: p(0).theta_qt(q=1,t=3)\n 0\n sage: p([2,1]).theta_qt(q=2,t=3)\n 3/16*p[2, 1]\n sage: s = p.realization_of().schur()\n sage: s([3]).theta_qt(q=0)*(1-t)*(1-t^2)*(1-t^3)\n t^3*s[1, 1, 1] + (t^2+t)*s[2, 1] + s[3]\n sage: p(1).theta_qt()\n p[]\n "
parent = self.parent()
BR = parent.base_ring()
p = parent.realization_of().power()
p_self = p(self)
if (t is None):
if hasattr(parent, 't'):
t = parent.t
else:
t = BR(QQ['t'].gen())
if (q is None):
if hasattr(parent, 'q'):
q = parent.q
else:
q = BR(QQ['q'].gen())
res = p_self.map_item((lambda m, c: (m, BR((prod([((1 - (q ** k)) / (1 - (t ** k))) for k in m]) * c)))))
return parent(res)
|
def theta_qt(self, q=None, t=None):
"\n Return the image of ``self`` under the `q,t`-deformed theta\n endomorphism which sends `p_k` to `\\frac{1-q^k}{1-t^k} \\cdot p_k`\n for all positive integers `k`.\n\n In general, this is well-defined outside of the powersum basis only\n if the base ring is a `\\QQ`-algebra.\n\n INPUT:\n\n - ``q``, ``t`` -- parameters (default: ``None``, in which case 'q'\n and 't' are used)\n\n EXAMPLES::\n\n sage: QQqt = QQ['q,t'].fraction_field()\n sage: q,t = QQqt.gens()\n sage: p = SymmetricFunctions(QQqt).p()\n sage: p([2]).theta_qt(q,t)\n ((-q^2+1)/(-t^2+1))*p[2]\n sage: p([2,1]).theta_qt(q,t)\n ((q^3-q^2-q+1)/(t^3-t^2-t+1))*p[2, 1]\n sage: p(0).theta_qt(q=1,t=3)\n 0\n sage: p([2,1]).theta_qt(q=2,t=3)\n 3/16*p[2, 1]\n sage: s = p.realization_of().schur()\n sage: s([3]).theta_qt(q=0)*(1-t)*(1-t^2)*(1-t^3)\n t^3*s[1, 1, 1] + (t^2+t)*s[2, 1] + s[3]\n sage: p(1).theta_qt()\n p[]\n "
parent = self.parent()
BR = parent.base_ring()
p = parent.realization_of().power()
p_self = p(self)
if (t is None):
if hasattr(parent, 't'):
t = parent.t
else:
t = BR(QQ['t'].gen())
if (q is None):
if hasattr(parent, 'q'):
q = parent.q
else:
q = BR(QQ['q'].gen())
res = p_self.map_item((lambda m, c: (m, BR((prod([((1 - (q ** k)) / (1 - (t ** k))) for k in m]) * c)))))
return parent(res)<|docstring|>Return the image of ``self`` under the `q,t`-deformed theta
endomorphism which sends `p_k` to `\frac{1-q^k}{1-t^k} \cdot p_k`
for all positive integers `k`.
In general, this is well-defined outside of the powersum basis only
if the base ring is a `\QQ`-algebra.
INPUT:
- ``q``, ``t`` -- parameters (default: ``None``, in which case 'q'
and 't' are used)
EXAMPLES::
sage: QQqt = QQ['q,t'].fraction_field()
sage: q,t = QQqt.gens()
sage: p = SymmetricFunctions(QQqt).p()
sage: p([2]).theta_qt(q,t)
((-q^2+1)/(-t^2+1))*p[2]
sage: p([2,1]).theta_qt(q,t)
((q^3-q^2-q+1)/(t^3-t^2-t+1))*p[2, 1]
sage: p(0).theta_qt(q=1,t=3)
0
sage: p([2,1]).theta_qt(q=2,t=3)
3/16*p[2, 1]
sage: s = p.realization_of().schur()
sage: s([3]).theta_qt(q=0)*(1-t)*(1-t^2)*(1-t^3)
t^3*s[1, 1, 1] + (t^2+t)*s[2, 1] + s[3]
sage: p(1).theta_qt()
p[]<|endoftext|>
|
04ad706e01d532725f66dfd7d0c4d0c1fd1e941bddce1d2005c3213ab26e8e9f
|
def omega_qt(self, q=None, t=None):
"\n Return the image of ``self`` under the `q,t`-deformed omega\n automorphism which sends `p_k` to\n `(-1)^{k-1} \\cdot \\frac{1-q^k}{1-t^k} \\cdot p_k` for all positive\n integers `k`.\n\n In general, this is well-defined outside of the powersum basis only\n if the base ring is a `\\QQ`-algebra.\n\n INPUT:\n\n - ``q``, ``t`` -- parameters (default: ``None``, in which case\n ``'q'`` and ``'t'`` are used)\n\n EXAMPLES::\n\n sage: QQqt = QQ['q,t'].fraction_field()\n sage: q,t = QQqt.gens()\n sage: p = SymmetricFunctions(QQqt).p()\n sage: p[5].omega_qt()\n ((-q^5+1)/(-t^5+1))*p[5]\n sage: p[5].omega_qt(q,t)\n ((-q^5+1)/(-t^5+1))*p[5]\n sage: p([2]).omega_qt(q,t)\n ((q^2-1)/(-t^2+1))*p[2]\n sage: p([2,1]).omega_qt(q,t)\n ((-q^3+q^2+q-1)/(t^3-t^2-t+1))*p[2, 1]\n sage: p([3,2]).omega_qt(5,q)\n -(2976/(q^5-q^3-q^2+1))*p[3, 2]\n sage: p(0).omega_qt()\n 0\n sage: p(1).omega_qt()\n p[]\n sage: H = SymmetricFunctions(QQqt).macdonald().H()\n sage: H([1,1]).omega_qt()\n ((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]\n sage: H([1,1]).omega_qt(q,t)\n ((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]\n sage: H([1,1]).omega_qt(t,q)\n ((t^3-t^2-t+1)/(q^3-q^2-q+1))*McdH[2]\n sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))\n sage: S = Sym.macdonald().S()\n sage: S([1,1]).omega_qt()\n ((q^2-q*t-q+t)/(t^3-t^2-t+1))*McdS[1, 1] + ((-q^2*t+q*t+q-1)/(-t^3+t^2+t-1))*McdS[2]\n sage: s = Sym.schur()\n sage: s(S([1,1]).omega_qt())\n s[2]\n "
parent = self.parent()
BR = parent.base_ring()
p = parent.realization_of().power()
p_self = p(self)
if (t is None):
if hasattr(parent, 't'):
t = parent.t
else:
t = BR(QQ['t'].gen())
if (q is None):
if hasattr(parent, 'q'):
q = parent.q
else:
q = BR(QQ['q'].gen())
f = (lambda part: prod([((((- 1) ** (i - 1)) * (1 - (q ** i))) / (1 - (t ** i))) for i in part]))
res = p_self.map_item((lambda m, c: (m, BR((f(m) * c)))))
return parent(res)
|
Return the image of ``self`` under the `q,t`-deformed omega
automorphism which sends `p_k` to
`(-1)^{k-1} \cdot \frac{1-q^k}{1-t^k} \cdot p_k` for all positive
integers `k`.
In general, this is well-defined outside of the powersum basis only
if the base ring is a `\QQ`-algebra.
INPUT:
- ``q``, ``t`` -- parameters (default: ``None``, in which case
``'q'`` and ``'t'`` are used)
EXAMPLES::
sage: QQqt = QQ['q,t'].fraction_field()
sage: q,t = QQqt.gens()
sage: p = SymmetricFunctions(QQqt).p()
sage: p[5].omega_qt()
((-q^5+1)/(-t^5+1))*p[5]
sage: p[5].omega_qt(q,t)
((-q^5+1)/(-t^5+1))*p[5]
sage: p([2]).omega_qt(q,t)
((q^2-1)/(-t^2+1))*p[2]
sage: p([2,1]).omega_qt(q,t)
((-q^3+q^2+q-1)/(t^3-t^2-t+1))*p[2, 1]
sage: p([3,2]).omega_qt(5,q)
-(2976/(q^5-q^3-q^2+1))*p[3, 2]
sage: p(0).omega_qt()
0
sage: p(1).omega_qt()
p[]
sage: H = SymmetricFunctions(QQqt).macdonald().H()
sage: H([1,1]).omega_qt()
((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]
sage: H([1,1]).omega_qt(q,t)
((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]
sage: H([1,1]).omega_qt(t,q)
((t^3-t^2-t+1)/(q^3-q^2-q+1))*McdH[2]
sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: S = Sym.macdonald().S()
sage: S([1,1]).omega_qt()
((q^2-q*t-q+t)/(t^3-t^2-t+1))*McdS[1, 1] + ((-q^2*t+q*t+q-1)/(-t^3+t^2+t-1))*McdS[2]
sage: s = Sym.schur()
sage: s(S([1,1]).omega_qt())
s[2]
|
src/sage/combinat/sf/sfa.py
|
omega_qt
|
bopopescu/sagesmc
| 5 |
python
|
def omega_qt(self, q=None, t=None):
"\n Return the image of ``self`` under the `q,t`-deformed omega\n automorphism which sends `p_k` to\n `(-1)^{k-1} \\cdot \\frac{1-q^k}{1-t^k} \\cdot p_k` for all positive\n integers `k`.\n\n In general, this is well-defined outside of the powersum basis only\n if the base ring is a `\\QQ`-algebra.\n\n INPUT:\n\n - ``q``, ``t`` -- parameters (default: ``None``, in which case\n ``'q'`` and ``'t'`` are used)\n\n EXAMPLES::\n\n sage: QQqt = QQ['q,t'].fraction_field()\n sage: q,t = QQqt.gens()\n sage: p = SymmetricFunctions(QQqt).p()\n sage: p[5].omega_qt()\n ((-q^5+1)/(-t^5+1))*p[5]\n sage: p[5].omega_qt(q,t)\n ((-q^5+1)/(-t^5+1))*p[5]\n sage: p([2]).omega_qt(q,t)\n ((q^2-1)/(-t^2+1))*p[2]\n sage: p([2,1]).omega_qt(q,t)\n ((-q^3+q^2+q-1)/(t^3-t^2-t+1))*p[2, 1]\n sage: p([3,2]).omega_qt(5,q)\n -(2976/(q^5-q^3-q^2+1))*p[3, 2]\n sage: p(0).omega_qt()\n 0\n sage: p(1).omega_qt()\n p[]\n sage: H = SymmetricFunctions(QQqt).macdonald().H()\n sage: H([1,1]).omega_qt()\n ((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]\n sage: H([1,1]).omega_qt(q,t)\n ((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]\n sage: H([1,1]).omega_qt(t,q)\n ((t^3-t^2-t+1)/(q^3-q^2-q+1))*McdH[2]\n sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))\n sage: S = Sym.macdonald().S()\n sage: S([1,1]).omega_qt()\n ((q^2-q*t-q+t)/(t^3-t^2-t+1))*McdS[1, 1] + ((-q^2*t+q*t+q-1)/(-t^3+t^2+t-1))*McdS[2]\n sage: s = Sym.schur()\n sage: s(S([1,1]).omega_qt())\n s[2]\n "
parent = self.parent()
BR = parent.base_ring()
p = parent.realization_of().power()
p_self = p(self)
if (t is None):
if hasattr(parent, 't'):
t = parent.t
else:
t = BR(QQ['t'].gen())
if (q is None):
if hasattr(parent, 'q'):
q = parent.q
else:
q = BR(QQ['q'].gen())
f = (lambda part: prod([((((- 1) ** (i - 1)) * (1 - (q ** i))) / (1 - (t ** i))) for i in part]))
res = p_self.map_item((lambda m, c: (m, BR((f(m) * c)))))
return parent(res)
|
def omega_qt(self, q=None, t=None):
"\n Return the image of ``self`` under the `q,t`-deformed omega\n automorphism which sends `p_k` to\n `(-1)^{k-1} \\cdot \\frac{1-q^k}{1-t^k} \\cdot p_k` for all positive\n integers `k`.\n\n In general, this is well-defined outside of the powersum basis only\n if the base ring is a `\\QQ`-algebra.\n\n INPUT:\n\n - ``q``, ``t`` -- parameters (default: ``None``, in which case\n ``'q'`` and ``'t'`` are used)\n\n EXAMPLES::\n\n sage: QQqt = QQ['q,t'].fraction_field()\n sage: q,t = QQqt.gens()\n sage: p = SymmetricFunctions(QQqt).p()\n sage: p[5].omega_qt()\n ((-q^5+1)/(-t^5+1))*p[5]\n sage: p[5].omega_qt(q,t)\n ((-q^5+1)/(-t^5+1))*p[5]\n sage: p([2]).omega_qt(q,t)\n ((q^2-1)/(-t^2+1))*p[2]\n sage: p([2,1]).omega_qt(q,t)\n ((-q^3+q^2+q-1)/(t^3-t^2-t+1))*p[2, 1]\n sage: p([3,2]).omega_qt(5,q)\n -(2976/(q^5-q^3-q^2+1))*p[3, 2]\n sage: p(0).omega_qt()\n 0\n sage: p(1).omega_qt()\n p[]\n sage: H = SymmetricFunctions(QQqt).macdonald().H()\n sage: H([1,1]).omega_qt()\n ((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]\n sage: H([1,1]).omega_qt(q,t)\n ((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]\n sage: H([1,1]).omega_qt(t,q)\n ((t^3-t^2-t+1)/(q^3-q^2-q+1))*McdH[2]\n sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))\n sage: S = Sym.macdonald().S()\n sage: S([1,1]).omega_qt()\n ((q^2-q*t-q+t)/(t^3-t^2-t+1))*McdS[1, 1] + ((-q^2*t+q*t+q-1)/(-t^3+t^2+t-1))*McdS[2]\n sage: s = Sym.schur()\n sage: s(S([1,1]).omega_qt())\n s[2]\n "
parent = self.parent()
BR = parent.base_ring()
p = parent.realization_of().power()
p_self = p(self)
if (t is None):
if hasattr(parent, 't'):
t = parent.t
else:
t = BR(QQ['t'].gen())
if (q is None):
if hasattr(parent, 'q'):
q = parent.q
else:
q = BR(QQ['q'].gen())
f = (lambda part: prod([((((- 1) ** (i - 1)) * (1 - (q ** i))) / (1 - (t ** i))) for i in part]))
res = p_self.map_item((lambda m, c: (m, BR((f(m) * c)))))
return parent(res)<|docstring|>Return the image of ``self`` under the `q,t`-deformed omega
automorphism which sends `p_k` to
`(-1)^{k-1} \cdot \frac{1-q^k}{1-t^k} \cdot p_k` for all positive
integers `k`.
In general, this is well-defined outside of the powersum basis only
if the base ring is a `\QQ`-algebra.
INPUT:
- ``q``, ``t`` -- parameters (default: ``None``, in which case
``'q'`` and ``'t'`` are used)
EXAMPLES::
sage: QQqt = QQ['q,t'].fraction_field()
sage: q,t = QQqt.gens()
sage: p = SymmetricFunctions(QQqt).p()
sage: p[5].omega_qt()
((-q^5+1)/(-t^5+1))*p[5]
sage: p[5].omega_qt(q,t)
((-q^5+1)/(-t^5+1))*p[5]
sage: p([2]).omega_qt(q,t)
((q^2-1)/(-t^2+1))*p[2]
sage: p([2,1]).omega_qt(q,t)
((-q^3+q^2+q-1)/(t^3-t^2-t+1))*p[2, 1]
sage: p([3,2]).omega_qt(5,q)
-(2976/(q^5-q^3-q^2+1))*p[3, 2]
sage: p(0).omega_qt()
0
sage: p(1).omega_qt()
p[]
sage: H = SymmetricFunctions(QQqt).macdonald().H()
sage: H([1,1]).omega_qt()
((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]
sage: H([1,1]).omega_qt(q,t)
((2*q^2-2*q*t-2*q+2*t)/(t^3-t^2-t+1))*McdH[1, 1] + ((q-1)/(t-1))*McdH[2]
sage: H([1,1]).omega_qt(t,q)
((t^3-t^2-t+1)/(q^3-q^2-q+1))*McdH[2]
sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: S = Sym.macdonald().S()
sage: S([1,1]).omega_qt()
((q^2-q*t-q+t)/(t^3-t^2-t+1))*McdS[1, 1] + ((-q^2*t+q*t+q-1)/(-t^3+t^2+t-1))*McdS[2]
sage: s = Sym.schur()
sage: s(S([1,1]).omega_qt())
s[2]<|endoftext|>
|
d97af1ea0777b2be914c6749fe9df924504d6a352068e40b7d464652ec8f7aa7
|
def itensor(self, x):
'\n Return the internal (tensor) product of ``self`` and ``x`` in the\n basis of ``self``.\n\n The internal tensor product can be defined as the linear extension\n of the definition on power sums\n `p_{\\lambda} \\ast p_{\\mu} = \\delta_{\\lambda,\\mu} z_{\\lambda}\n p_{\\lambda}`, where `z_{\\lambda} = (1^{r_1} r_1!) (2^{r_2} r_2!)\n \\cdots` for `\\lambda = (1^{r_1} 2^{r_2} \\cdots )` and where `\\ast`\n denotes the internal tensor product.\n The internal tensor product is also known as the Kronecker product,\n or as the second multiplication on the ring of symmetric functions.\n\n Note that the internal product of any two homogeneous symmetric\n functions of equal degrees is a homogeneous symmetric function of the\n same degree. On the other hand, the internal product of two homogeneous\n symmetric functions of distinct degrees is `0`.\n\n .. NOTE::\n\n The internal product is sometimes referred to as "inner product"\n in the literature, but unfortunately this name is shared by a\n different operation, namely the Hall inner product\n (see :meth:`scalar`).\n\n INPUT:\n\n - ``x`` -- element of the ring of symmetric functions over the\n same base ring as ``self``\n\n OUTPUT:\n\n - the internal product of ``self`` with ``x`` (an element of the\n ring of symmetric functions in the same basis as ``self``)\n\n The methods :meth:`itensor`, :meth:`internal_product`,\n :meth:`kronecker_product`, :meth:`inner_tensor` are all\n synonyms.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([2,1])\n sage: b = s([3])\n sage: a.itensor(b)\n s[2, 1]\n sage: c = s([3,2,1])\n sage: c.itensor(c)\n s[1, 1, 1, 1, 1, 1] + 2*s[2, 1, 1, 1, 1] + 3*s[2, 2, 1, 1] + 2*s[2, 2, 2] + 4*s[3, 1, 1, 1] + 5*s[3, 2, 1] + 2*s[3, 3] + 4*s[4, 1, 1] + 3*s[4, 2] + 2*s[5, 1] + s[6]\n\n There are few quantitative results pertaining to Kronecker products\n in general, which makes their computation so difficult. Let us test\n a few of them in different bases.\n\n The Kronecker product of any homogeneous symmetric function `f` of\n degree `n` with the `n`-th complete homogeneous symmetric function\n ``h[n]`` (a.k.a. ``s[n]``) is `f`::\n\n sage: h = SymmetricFunctions(ZZ).h()\n sage: all( h([5]).itensor(h(p)) == h(p) for p in Partitions(5) )\n True\n\n The Kronecker product of a Schur function `s_{\\lambda}` with the `n`-th\n elementary symmetric function ``e[n]``, where `n = \\left| \\lambda\n \\right|`, is `s_{\\lambda\'}` (where `\\lambda\'` is the conjugate\n partition of `\\lambda`)::\n\n sage: F = CyclotomicField(12)\n sage: s = SymmetricFunctions(F).s()\n sage: e = SymmetricFunctions(F).e()\n sage: all( e([5]).itensor(s(p)) == s(p.conjugate()) for p in Partitions(5) )\n True\n\n The Kronecker product is commutative::\n\n sage: e = SymmetricFunctions(FiniteField(19)).e()\n sage: m = SymmetricFunctions(FiniteField(19)).m()\n sage: all( all( e(p).itensor(m(q)) == m(q).itensor(e(p)) for q in Partitions(4) )\n ....: for p in Partitions(4) )\n True\n\n sage: F = FractionField(QQ[\'q\',\'t\'])\n sage: mq = SymmetricFunctions(F).macdonald().Q()\n sage: mh = SymmetricFunctions(F).macdonald().H()\n sage: all( all( mq(p).itensor(mh(r)) == mh(r).itensor(mq(p)) for r in Partitions(4) )\n ....: for p in Partitions(3) ) # long time\n True\n\n Let us check (on examples) Proposition 5.2 of Gelfand, Krob, Lascoux, Leclerc,\n Retakh, Thibon, "Noncommutative symmetric functions", :arXiv:`hep-th/9407124`, for\n `r = 2`::\n\n sage: e = SymmetricFunctions(FiniteField(29)).e()\n sage: s = SymmetricFunctions(FiniteField(29)).s()\n sage: m = SymmetricFunctions(FiniteField(29)).m()\n sage: def tensor_copr(u, v, w): # computes \\mu ((u \\otimes v) * \\Delta(w)) with\n ....: # * meaning Kronecker product and \\mu meaning the\n ....: # usual multiplication.\n ....: result = w.parent().zero()\n ....: for partition_pair, coeff in w.coproduct().monomial_coefficients().iteritems():\n ....: result += coeff * w.parent()(u).itensor(partition_pair[0]) * w.parent()(v).itensor(partition_pair[1])\n ....: return result\n sage: all( all( all( tensor_copr(e[u], s[v], m[w]) == (e[u] * s[v]).itensor(m[w])\n ....: for w in Partitions(5) )\n ....: for v in Partitions(2) )\n ....: for u in Partitions(3) ) # long time\n True\n\n Some examples from Briand, Orellana, Rosas, "The stability of the Kronecker\n products of Schur functions." :arXiv:`0907.4652`::\n\n sage: s = SymmetricFunctions(ZZ).s()\n sage: s[2,2].itensor(s[2,2])\n s[1, 1, 1, 1] + s[2, 2] + s[4]\n sage: s[3,2].itensor(s[3,2])\n s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]\n sage: s[4,2].itensor(s[4,2])\n s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1] + s[6]\n\n An example from p. 220 of Thibon, "Hopf algebras of symmetric functions\n and tensor products of symmetric group representations", International\n Journal of Algebra and Computation, 0000:0000:0000:0000:0000:0000:0000:0000\n\n sage: s = SymmetricFunctions(QQbar).s()\n sage: s[2,1].itensor(s[2,1])\n s[1, 1, 1] + s[2, 1] + s[3]\n\n TESTS::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([8,8])\n sage: a.itensor(a)\n s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2] + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1] + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2] + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2] + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2] + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]\n sage: s[8].itensor(s[7])\n 0\n sage: s(0).itensor(s(0))\n 0\n sage: s(1).itensor(s(0))\n 0\n sage: s(0).itensor(s(1))\n 0\n sage: s(1).itensor(s(1))\n s[]\n\n Same over the ring of integers::\n\n sage: s = SymmetricFunctions(ZZ).s()\n sage: a = s([8,8])\n sage: a.itensor(a)\n s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2] + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1] + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2] + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2] + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2] + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]\n sage: s[8].itensor(s[7])\n 0\n sage: s(0).itensor(s(0))\n 0\n sage: s(1).itensor(s(0))\n 0\n sage: s(0).itensor(s(1))\n 0\n sage: s(1).itensor(s(1))\n s[]\n\n Theorem 2.1 in Bessenrodt, van Willigenburg, :arXiv:`1105.3170v2`::\n\n sage: s = SymmetricFunctions(ZZ).s()\n sage: all( all( max( r[0] for r in s(p).itensor(s(q)).monomial_coefficients().keys() )\n ....: == sum( min(p[i], q.get_part(i)) for i in range(len(p)) )\n ....: for p in Partitions(4) )\n ....: for q in Partitions(4) )\n True\n sage: all( all( max( len(r) for r in s(p).itensor(s(q)).monomial_coefficients().keys() )\n ....: == sum( min(p[i], q.conjugate().get_part(i)) for i in range(len(p)) )\n ....: for p in Partitions(4) )\n ....: for q in Partitions(4) )\n True\n\n Check that the basis and ground ring of ``self`` are preserved::\n\n sage: F = CyclotomicField(12)\n sage: s = SymmetricFunctions(F).s()\n sage: e = SymmetricFunctions(F).e()\n sage: e[3].itensor(s[3])\n e[3]\n sage: s[3].itensor(e[3])\n s[1, 1, 1]\n sage: parent(e[3].itensor(s[3]))\n Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis\n sage: parent(s[3].itensor(e[3]))\n Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the Schur basis\n\n .. NOTE::\n\n The currently existing implementation of this function is\n technically unsatisfactory. It distinguishes the case when the\n base ring is a `\\QQ`-algebra (in which case the Kronecker product\n can be easily computed using the power sum basis) from the case\n where it isn\'t. In the latter, it does a computation using\n universal coefficients, again distinguishing the case when it is\n able to compute the "corresponding" basis of the symmetric function\n algebra over `\\QQ` (using the ``corresponding_basis_over`` hack)\n from the case when it isn\'t (in which case it transforms everything\n into the Schur basis, which is slow).\n '
parent = self.parent()
if parent.has_coerce_map_from(QQ):
p = parent.realization_of().power()
f = (lambda part1, part2: (zee(part1) * p(part1)))
return parent(p._apply_multi_module_morphism(p(self), p(x), f, orthogonal=True))
else:
comp_parent = parent
comp_self = self
corresponding_parent_over_QQ = parent.corresponding_basis_over(QQ)
if (corresponding_parent_over_QQ is None):
comp_parent = parent.realization_of().schur()
comp_self = comp_parent(self)
from sage.combinat.sf.sf import SymmetricFunctions
corresponding_parent_over_QQ = SymmetricFunctions(QQ).schur()
comp_x = comp_parent(x)
result = comp_parent.zero()
for (lam, a) in comp_self.monomial_coefficients().items():
for (mu, b) in comp_x.monomial_coefficients().items():
lam_star_mu = corresponding_parent_over_QQ(lam).itensor(corresponding_parent_over_QQ(mu))
for (nu, c) in lam_star_mu.monomial_coefficients().items():
result += (((a * b) * comp_parent.base_ring()(c)) * comp_parent(nu))
return parent(result)
|
Return the internal (tensor) product of ``self`` and ``x`` in the
basis of ``self``.
The internal tensor product can be defined as the linear extension
of the definition on power sums
`p_{\lambda} \ast p_{\mu} = \delta_{\lambda,\mu} z_{\lambda}
p_{\lambda}`, where `z_{\lambda} = (1^{r_1} r_1!) (2^{r_2} r_2!)
\cdots` for `\lambda = (1^{r_1} 2^{r_2} \cdots )` and where `\ast`
denotes the internal tensor product.
The internal tensor product is also known as the Kronecker product,
or as the second multiplication on the ring of symmetric functions.
Note that the internal product of any two homogeneous symmetric
functions of equal degrees is a homogeneous symmetric function of the
same degree. On the other hand, the internal product of two homogeneous
symmetric functions of distinct degrees is `0`.
.. NOTE::
The internal product is sometimes referred to as "inner product"
in the literature, but unfortunately this name is shared by a
different operation, namely the Hall inner product
(see :meth:`scalar`).
INPUT:
- ``x`` -- element of the ring of symmetric functions over the
same base ring as ``self``
OUTPUT:
- the internal product of ``self`` with ``x`` (an element of the
ring of symmetric functions in the same basis as ``self``)
The methods :meth:`itensor`, :meth:`internal_product`,
:meth:`kronecker_product`, :meth:`inner_tensor` are all
synonyms.
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([2,1])
sage: b = s([3])
sage: a.itensor(b)
s[2, 1]
sage: c = s([3,2,1])
sage: c.itensor(c)
s[1, 1, 1, 1, 1, 1] + 2*s[2, 1, 1, 1, 1] + 3*s[2, 2, 1, 1] + 2*s[2, 2, 2] + 4*s[3, 1, 1, 1] + 5*s[3, 2, 1] + 2*s[3, 3] + 4*s[4, 1, 1] + 3*s[4, 2] + 2*s[5, 1] + s[6]
There are few quantitative results pertaining to Kronecker products
in general, which makes their computation so difficult. Let us test
a few of them in different bases.
The Kronecker product of any homogeneous symmetric function `f` of
degree `n` with the `n`-th complete homogeneous symmetric function
``h[n]`` (a.k.a. ``s[n]``) is `f`::
sage: h = SymmetricFunctions(ZZ).h()
sage: all( h([5]).itensor(h(p)) == h(p) for p in Partitions(5) )
True
The Kronecker product of a Schur function `s_{\lambda}` with the `n`-th
elementary symmetric function ``e[n]``, where `n = \left| \lambda
\right|`, is `s_{\lambda'}` (where `\lambda'` is the conjugate
partition of `\lambda`)::
sage: F = CyclotomicField(12)
sage: s = SymmetricFunctions(F).s()
sage: e = SymmetricFunctions(F).e()
sage: all( e([5]).itensor(s(p)) == s(p.conjugate()) for p in Partitions(5) )
True
The Kronecker product is commutative::
sage: e = SymmetricFunctions(FiniteField(19)).e()
sage: m = SymmetricFunctions(FiniteField(19)).m()
sage: all( all( e(p).itensor(m(q)) == m(q).itensor(e(p)) for q in Partitions(4) )
....: for p in Partitions(4) )
True
sage: F = FractionField(QQ['q','t'])
sage: mq = SymmetricFunctions(F).macdonald().Q()
sage: mh = SymmetricFunctions(F).macdonald().H()
sage: all( all( mq(p).itensor(mh(r)) == mh(r).itensor(mq(p)) for r in Partitions(4) )
....: for p in Partitions(3) ) # long time
True
Let us check (on examples) Proposition 5.2 of Gelfand, Krob, Lascoux, Leclerc,
Retakh, Thibon, "Noncommutative symmetric functions", :arXiv:`hep-th/9407124`, for
`r = 2`::
sage: e = SymmetricFunctions(FiniteField(29)).e()
sage: s = SymmetricFunctions(FiniteField(29)).s()
sage: m = SymmetricFunctions(FiniteField(29)).m()
sage: def tensor_copr(u, v, w): # computes \mu ((u \otimes v) * \Delta(w)) with
....: # * meaning Kronecker product and \mu meaning the
....: # usual multiplication.
....: result = w.parent().zero()
....: for partition_pair, coeff in w.coproduct().monomial_coefficients().iteritems():
....: result += coeff * w.parent()(u).itensor(partition_pair[0]) * w.parent()(v).itensor(partition_pair[1])
....: return result
sage: all( all( all( tensor_copr(e[u], s[v], m[w]) == (e[u] * s[v]).itensor(m[w])
....: for w in Partitions(5) )
....: for v in Partitions(2) )
....: for u in Partitions(3) ) # long time
True
Some examples from Briand, Orellana, Rosas, "The stability of the Kronecker
products of Schur functions." :arXiv:`0907.4652`::
sage: s = SymmetricFunctions(ZZ).s()
sage: s[2,2].itensor(s[2,2])
s[1, 1, 1, 1] + s[2, 2] + s[4]
sage: s[3,2].itensor(s[3,2])
s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]
sage: s[4,2].itensor(s[4,2])
s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1] + s[6]
An example from p. 220 of Thibon, "Hopf algebras of symmetric functions
and tensor products of symmetric group representations", International
Journal of Algebra and Computation, 0000:0000:0000:0000:0000:0000:0000:0000
sage: s = SymmetricFunctions(QQbar).s()
sage: s[2,1].itensor(s[2,1])
s[1, 1, 1] + s[2, 1] + s[3]
TESTS::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([8,8])
sage: a.itensor(a)
s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2] + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1] + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2] + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2] + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2] + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
sage: s[8].itensor(s[7])
0
sage: s(0).itensor(s(0))
0
sage: s(1).itensor(s(0))
0
sage: s(0).itensor(s(1))
0
sage: s(1).itensor(s(1))
s[]
Same over the ring of integers::
sage: s = SymmetricFunctions(ZZ).s()
sage: a = s([8,8])
sage: a.itensor(a)
s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2] + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1] + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2] + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2] + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2] + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
sage: s[8].itensor(s[7])
0
sage: s(0).itensor(s(0))
0
sage: s(1).itensor(s(0))
0
sage: s(0).itensor(s(1))
0
sage: s(1).itensor(s(1))
s[]
Theorem 2.1 in Bessenrodt, van Willigenburg, :arXiv:`1105.3170v2`::
sage: s = SymmetricFunctions(ZZ).s()
sage: all( all( max( r[0] for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
....: == sum( min(p[i], q.get_part(i)) for i in range(len(p)) )
....: for p in Partitions(4) )
....: for q in Partitions(4) )
True
sage: all( all( max( len(r) for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
....: == sum( min(p[i], q.conjugate().get_part(i)) for i in range(len(p)) )
....: for p in Partitions(4) )
....: for q in Partitions(4) )
True
Check that the basis and ground ring of ``self`` are preserved::
sage: F = CyclotomicField(12)
sage: s = SymmetricFunctions(F).s()
sage: e = SymmetricFunctions(F).e()
sage: e[3].itensor(s[3])
e[3]
sage: s[3].itensor(e[3])
s[1, 1, 1]
sage: parent(e[3].itensor(s[3]))
Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis
sage: parent(s[3].itensor(e[3]))
Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the Schur basis
.. NOTE::
The currently existing implementation of this function is
technically unsatisfactory. It distinguishes the case when the
base ring is a `\QQ`-algebra (in which case the Kronecker product
can be easily computed using the power sum basis) from the case
where it isn't. In the latter, it does a computation using
universal coefficients, again distinguishing the case when it is
able to compute the "corresponding" basis of the symmetric function
algebra over `\QQ` (using the ``corresponding_basis_over`` hack)
from the case when it isn't (in which case it transforms everything
into the Schur basis, which is slow).
|
src/sage/combinat/sf/sfa.py
|
itensor
|
bopopescu/sagesmc
| 5 |
python
|
def itensor(self, x):
'\n Return the internal (tensor) product of ``self`` and ``x`` in the\n basis of ``self``.\n\n The internal tensor product can be defined as the linear extension\n of the definition on power sums\n `p_{\\lambda} \\ast p_{\\mu} = \\delta_{\\lambda,\\mu} z_{\\lambda}\n p_{\\lambda}`, where `z_{\\lambda} = (1^{r_1} r_1!) (2^{r_2} r_2!)\n \\cdots` for `\\lambda = (1^{r_1} 2^{r_2} \\cdots )` and where `\\ast`\n denotes the internal tensor product.\n The internal tensor product is also known as the Kronecker product,\n or as the second multiplication on the ring of symmetric functions.\n\n Note that the internal product of any two homogeneous symmetric\n functions of equal degrees is a homogeneous symmetric function of the\n same degree. On the other hand, the internal product of two homogeneous\n symmetric functions of distinct degrees is `0`.\n\n .. NOTE::\n\n The internal product is sometimes referred to as "inner product"\n in the literature, but unfortunately this name is shared by a\n different operation, namely the Hall inner product\n (see :meth:`scalar`).\n\n INPUT:\n\n - ``x`` -- element of the ring of symmetric functions over the\n same base ring as ``self``\n\n OUTPUT:\n\n - the internal product of ``self`` with ``x`` (an element of the\n ring of symmetric functions in the same basis as ``self``)\n\n The methods :meth:`itensor`, :meth:`internal_product`,\n :meth:`kronecker_product`, :meth:`inner_tensor` are all\n synonyms.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([2,1])\n sage: b = s([3])\n sage: a.itensor(b)\n s[2, 1]\n sage: c = s([3,2,1])\n sage: c.itensor(c)\n s[1, 1, 1, 1, 1, 1] + 2*s[2, 1, 1, 1, 1] + 3*s[2, 2, 1, 1] + 2*s[2, 2, 2] + 4*s[3, 1, 1, 1] + 5*s[3, 2, 1] + 2*s[3, 3] + 4*s[4, 1, 1] + 3*s[4, 2] + 2*s[5, 1] + s[6]\n\n There are few quantitative results pertaining to Kronecker products\n in general, which makes their computation so difficult. Let us test\n a few of them in different bases.\n\n The Kronecker product of any homogeneous symmetric function `f` of\n degree `n` with the `n`-th complete homogeneous symmetric function\n ``h[n]`` (a.k.a. ``s[n]``) is `f`::\n\n sage: h = SymmetricFunctions(ZZ).h()\n sage: all( h([5]).itensor(h(p)) == h(p) for p in Partitions(5) )\n True\n\n The Kronecker product of a Schur function `s_{\\lambda}` with the `n`-th\n elementary symmetric function ``e[n]``, where `n = \\left| \\lambda\n \\right|`, is `s_{\\lambda\'}` (where `\\lambda\'` is the conjugate\n partition of `\\lambda`)::\n\n sage: F = CyclotomicField(12)\n sage: s = SymmetricFunctions(F).s()\n sage: e = SymmetricFunctions(F).e()\n sage: all( e([5]).itensor(s(p)) == s(p.conjugate()) for p in Partitions(5) )\n True\n\n The Kronecker product is commutative::\n\n sage: e = SymmetricFunctions(FiniteField(19)).e()\n sage: m = SymmetricFunctions(FiniteField(19)).m()\n sage: all( all( e(p).itensor(m(q)) == m(q).itensor(e(p)) for q in Partitions(4) )\n ....: for p in Partitions(4) )\n True\n\n sage: F = FractionField(QQ[\'q\',\'t\'])\n sage: mq = SymmetricFunctions(F).macdonald().Q()\n sage: mh = SymmetricFunctions(F).macdonald().H()\n sage: all( all( mq(p).itensor(mh(r)) == mh(r).itensor(mq(p)) for r in Partitions(4) )\n ....: for p in Partitions(3) ) # long time\n True\n\n Let us check (on examples) Proposition 5.2 of Gelfand, Krob, Lascoux, Leclerc,\n Retakh, Thibon, "Noncommutative symmetric functions", :arXiv:`hep-th/9407124`, for\n `r = 2`::\n\n sage: e = SymmetricFunctions(FiniteField(29)).e()\n sage: s = SymmetricFunctions(FiniteField(29)).s()\n sage: m = SymmetricFunctions(FiniteField(29)).m()\n sage: def tensor_copr(u, v, w): # computes \\mu ((u \\otimes v) * \\Delta(w)) with\n ....: # * meaning Kronecker product and \\mu meaning the\n ....: # usual multiplication.\n ....: result = w.parent().zero()\n ....: for partition_pair, coeff in w.coproduct().monomial_coefficients().iteritems():\n ....: result += coeff * w.parent()(u).itensor(partition_pair[0]) * w.parent()(v).itensor(partition_pair[1])\n ....: return result\n sage: all( all( all( tensor_copr(e[u], s[v], m[w]) == (e[u] * s[v]).itensor(m[w])\n ....: for w in Partitions(5) )\n ....: for v in Partitions(2) )\n ....: for u in Partitions(3) ) # long time\n True\n\n Some examples from Briand, Orellana, Rosas, "The stability of the Kronecker\n products of Schur functions." :arXiv:`0907.4652`::\n\n sage: s = SymmetricFunctions(ZZ).s()\n sage: s[2,2].itensor(s[2,2])\n s[1, 1, 1, 1] + s[2, 2] + s[4]\n sage: s[3,2].itensor(s[3,2])\n s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]\n sage: s[4,2].itensor(s[4,2])\n s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1] + s[6]\n\n An example from p. 220 of Thibon, "Hopf algebras of symmetric functions\n and tensor products of symmetric group representations", International\n Journal of Algebra and Computation, 0000:0000:0000:0000:0000:0000:0000:0000\n\n sage: s = SymmetricFunctions(QQbar).s()\n sage: s[2,1].itensor(s[2,1])\n s[1, 1, 1] + s[2, 1] + s[3]\n\n TESTS::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([8,8])\n sage: a.itensor(a)\n s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2] + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1] + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2] + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2] + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2] + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]\n sage: s[8].itensor(s[7])\n 0\n sage: s(0).itensor(s(0))\n 0\n sage: s(1).itensor(s(0))\n 0\n sage: s(0).itensor(s(1))\n 0\n sage: s(1).itensor(s(1))\n s[]\n\n Same over the ring of integers::\n\n sage: s = SymmetricFunctions(ZZ).s()\n sage: a = s([8,8])\n sage: a.itensor(a)\n s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2] + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1] + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2] + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2] + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2] + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]\n sage: s[8].itensor(s[7])\n 0\n sage: s(0).itensor(s(0))\n 0\n sage: s(1).itensor(s(0))\n 0\n sage: s(0).itensor(s(1))\n 0\n sage: s(1).itensor(s(1))\n s[]\n\n Theorem 2.1 in Bessenrodt, van Willigenburg, :arXiv:`1105.3170v2`::\n\n sage: s = SymmetricFunctions(ZZ).s()\n sage: all( all( max( r[0] for r in s(p).itensor(s(q)).monomial_coefficients().keys() )\n ....: == sum( min(p[i], q.get_part(i)) for i in range(len(p)) )\n ....: for p in Partitions(4) )\n ....: for q in Partitions(4) )\n True\n sage: all( all( max( len(r) for r in s(p).itensor(s(q)).monomial_coefficients().keys() )\n ....: == sum( min(p[i], q.conjugate().get_part(i)) for i in range(len(p)) )\n ....: for p in Partitions(4) )\n ....: for q in Partitions(4) )\n True\n\n Check that the basis and ground ring of ``self`` are preserved::\n\n sage: F = CyclotomicField(12)\n sage: s = SymmetricFunctions(F).s()\n sage: e = SymmetricFunctions(F).e()\n sage: e[3].itensor(s[3])\n e[3]\n sage: s[3].itensor(e[3])\n s[1, 1, 1]\n sage: parent(e[3].itensor(s[3]))\n Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis\n sage: parent(s[3].itensor(e[3]))\n Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the Schur basis\n\n .. NOTE::\n\n The currently existing implementation of this function is\n technically unsatisfactory. It distinguishes the case when the\n base ring is a `\\QQ`-algebra (in which case the Kronecker product\n can be easily computed using the power sum basis) from the case\n where it isn\'t. In the latter, it does a computation using\n universal coefficients, again distinguishing the case when it is\n able to compute the "corresponding" basis of the symmetric function\n algebra over `\\QQ` (using the ``corresponding_basis_over`` hack)\n from the case when it isn\'t (in which case it transforms everything\n into the Schur basis, which is slow).\n '
parent = self.parent()
if parent.has_coerce_map_from(QQ):
p = parent.realization_of().power()
f = (lambda part1, part2: (zee(part1) * p(part1)))
return parent(p._apply_multi_module_morphism(p(self), p(x), f, orthogonal=True))
else:
comp_parent = parent
comp_self = self
corresponding_parent_over_QQ = parent.corresponding_basis_over(QQ)
if (corresponding_parent_over_QQ is None):
comp_parent = parent.realization_of().schur()
comp_self = comp_parent(self)
from sage.combinat.sf.sf import SymmetricFunctions
corresponding_parent_over_QQ = SymmetricFunctions(QQ).schur()
comp_x = comp_parent(x)
result = comp_parent.zero()
for (lam, a) in comp_self.monomial_coefficients().items():
for (mu, b) in comp_x.monomial_coefficients().items():
lam_star_mu = corresponding_parent_over_QQ(lam).itensor(corresponding_parent_over_QQ(mu))
for (nu, c) in lam_star_mu.monomial_coefficients().items():
result += (((a * b) * comp_parent.base_ring()(c)) * comp_parent(nu))
return parent(result)
|
def itensor(self, x):
'\n Return the internal (tensor) product of ``self`` and ``x`` in the\n basis of ``self``.\n\n The internal tensor product can be defined as the linear extension\n of the definition on power sums\n `p_{\\lambda} \\ast p_{\\mu} = \\delta_{\\lambda,\\mu} z_{\\lambda}\n p_{\\lambda}`, where `z_{\\lambda} = (1^{r_1} r_1!) (2^{r_2} r_2!)\n \\cdots` for `\\lambda = (1^{r_1} 2^{r_2} \\cdots )` and where `\\ast`\n denotes the internal tensor product.\n The internal tensor product is also known as the Kronecker product,\n or as the second multiplication on the ring of symmetric functions.\n\n Note that the internal product of any two homogeneous symmetric\n functions of equal degrees is a homogeneous symmetric function of the\n same degree. On the other hand, the internal product of two homogeneous\n symmetric functions of distinct degrees is `0`.\n\n .. NOTE::\n\n The internal product is sometimes referred to as "inner product"\n in the literature, but unfortunately this name is shared by a\n different operation, namely the Hall inner product\n (see :meth:`scalar`).\n\n INPUT:\n\n - ``x`` -- element of the ring of symmetric functions over the\n same base ring as ``self``\n\n OUTPUT:\n\n - the internal product of ``self`` with ``x`` (an element of the\n ring of symmetric functions in the same basis as ``self``)\n\n The methods :meth:`itensor`, :meth:`internal_product`,\n :meth:`kronecker_product`, :meth:`inner_tensor` are all\n synonyms.\n\n EXAMPLES::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([2,1])\n sage: b = s([3])\n sage: a.itensor(b)\n s[2, 1]\n sage: c = s([3,2,1])\n sage: c.itensor(c)\n s[1, 1, 1, 1, 1, 1] + 2*s[2, 1, 1, 1, 1] + 3*s[2, 2, 1, 1] + 2*s[2, 2, 2] + 4*s[3, 1, 1, 1] + 5*s[3, 2, 1] + 2*s[3, 3] + 4*s[4, 1, 1] + 3*s[4, 2] + 2*s[5, 1] + s[6]\n\n There are few quantitative results pertaining to Kronecker products\n in general, which makes their computation so difficult. Let us test\n a few of them in different bases.\n\n The Kronecker product of any homogeneous symmetric function `f` of\n degree `n` with the `n`-th complete homogeneous symmetric function\n ``h[n]`` (a.k.a. ``s[n]``) is `f`::\n\n sage: h = SymmetricFunctions(ZZ).h()\n sage: all( h([5]).itensor(h(p)) == h(p) for p in Partitions(5) )\n True\n\n The Kronecker product of a Schur function `s_{\\lambda}` with the `n`-th\n elementary symmetric function ``e[n]``, where `n = \\left| \\lambda\n \\right|`, is `s_{\\lambda\'}` (where `\\lambda\'` is the conjugate\n partition of `\\lambda`)::\n\n sage: F = CyclotomicField(12)\n sage: s = SymmetricFunctions(F).s()\n sage: e = SymmetricFunctions(F).e()\n sage: all( e([5]).itensor(s(p)) == s(p.conjugate()) for p in Partitions(5) )\n True\n\n The Kronecker product is commutative::\n\n sage: e = SymmetricFunctions(FiniteField(19)).e()\n sage: m = SymmetricFunctions(FiniteField(19)).m()\n sage: all( all( e(p).itensor(m(q)) == m(q).itensor(e(p)) for q in Partitions(4) )\n ....: for p in Partitions(4) )\n True\n\n sage: F = FractionField(QQ[\'q\',\'t\'])\n sage: mq = SymmetricFunctions(F).macdonald().Q()\n sage: mh = SymmetricFunctions(F).macdonald().H()\n sage: all( all( mq(p).itensor(mh(r)) == mh(r).itensor(mq(p)) for r in Partitions(4) )\n ....: for p in Partitions(3) ) # long time\n True\n\n Let us check (on examples) Proposition 5.2 of Gelfand, Krob, Lascoux, Leclerc,\n Retakh, Thibon, "Noncommutative symmetric functions", :arXiv:`hep-th/9407124`, for\n `r = 2`::\n\n sage: e = SymmetricFunctions(FiniteField(29)).e()\n sage: s = SymmetricFunctions(FiniteField(29)).s()\n sage: m = SymmetricFunctions(FiniteField(29)).m()\n sage: def tensor_copr(u, v, w): # computes \\mu ((u \\otimes v) * \\Delta(w)) with\n ....: # * meaning Kronecker product and \\mu meaning the\n ....: # usual multiplication.\n ....: result = w.parent().zero()\n ....: for partition_pair, coeff in w.coproduct().monomial_coefficients().iteritems():\n ....: result += coeff * w.parent()(u).itensor(partition_pair[0]) * w.parent()(v).itensor(partition_pair[1])\n ....: return result\n sage: all( all( all( tensor_copr(e[u], s[v], m[w]) == (e[u] * s[v]).itensor(m[w])\n ....: for w in Partitions(5) )\n ....: for v in Partitions(2) )\n ....: for u in Partitions(3) ) # long time\n True\n\n Some examples from Briand, Orellana, Rosas, "The stability of the Kronecker\n products of Schur functions." :arXiv:`0907.4652`::\n\n sage: s = SymmetricFunctions(ZZ).s()\n sage: s[2,2].itensor(s[2,2])\n s[1, 1, 1, 1] + s[2, 2] + s[4]\n sage: s[3,2].itensor(s[3,2])\n s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]\n sage: s[4,2].itensor(s[4,2])\n s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1] + s[6]\n\n An example from p. 220 of Thibon, "Hopf algebras of symmetric functions\n and tensor products of symmetric group representations", International\n Journal of Algebra and Computation, 0000:0000:0000:0000:0000:0000:0000:0000\n\n sage: s = SymmetricFunctions(QQbar).s()\n sage: s[2,1].itensor(s[2,1])\n s[1, 1, 1] + s[2, 1] + s[3]\n\n TESTS::\n\n sage: s = SymmetricFunctions(QQ).s()\n sage: a = s([8,8])\n sage: a.itensor(a)\n s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2] + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1] + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2] + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2] + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2] + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]\n sage: s[8].itensor(s[7])\n 0\n sage: s(0).itensor(s(0))\n 0\n sage: s(1).itensor(s(0))\n 0\n sage: s(0).itensor(s(1))\n 0\n sage: s(1).itensor(s(1))\n s[]\n\n Same over the ring of integers::\n\n sage: s = SymmetricFunctions(ZZ).s()\n sage: a = s([8,8])\n sage: a.itensor(a)\n s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2] + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1] + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2] + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2] + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2] + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]\n sage: s[8].itensor(s[7])\n 0\n sage: s(0).itensor(s(0))\n 0\n sage: s(1).itensor(s(0))\n 0\n sage: s(0).itensor(s(1))\n 0\n sage: s(1).itensor(s(1))\n s[]\n\n Theorem 2.1 in Bessenrodt, van Willigenburg, :arXiv:`1105.3170v2`::\n\n sage: s = SymmetricFunctions(ZZ).s()\n sage: all( all( max( r[0] for r in s(p).itensor(s(q)).monomial_coefficients().keys() )\n ....: == sum( min(p[i], q.get_part(i)) for i in range(len(p)) )\n ....: for p in Partitions(4) )\n ....: for q in Partitions(4) )\n True\n sage: all( all( max( len(r) for r in s(p).itensor(s(q)).monomial_coefficients().keys() )\n ....: == sum( min(p[i], q.conjugate().get_part(i)) for i in range(len(p)) )\n ....: for p in Partitions(4) )\n ....: for q in Partitions(4) )\n True\n\n Check that the basis and ground ring of ``self`` are preserved::\n\n sage: F = CyclotomicField(12)\n sage: s = SymmetricFunctions(F).s()\n sage: e = SymmetricFunctions(F).e()\n sage: e[3].itensor(s[3])\n e[3]\n sage: s[3].itensor(e[3])\n s[1, 1, 1]\n sage: parent(e[3].itensor(s[3]))\n Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis\n sage: parent(s[3].itensor(e[3]))\n Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the Schur basis\n\n .. NOTE::\n\n The currently existing implementation of this function is\n technically unsatisfactory. It distinguishes the case when the\n base ring is a `\\QQ`-algebra (in which case the Kronecker product\n can be easily computed using the power sum basis) from the case\n where it isn\'t. In the latter, it does a computation using\n universal coefficients, again distinguishing the case when it is\n able to compute the "corresponding" basis of the symmetric function\n algebra over `\\QQ` (using the ``corresponding_basis_over`` hack)\n from the case when it isn\'t (in which case it transforms everything\n into the Schur basis, which is slow).\n '
parent = self.parent()
if parent.has_coerce_map_from(QQ):
p = parent.realization_of().power()
f = (lambda part1, part2: (zee(part1) * p(part1)))
return parent(p._apply_multi_module_morphism(p(self), p(x), f, orthogonal=True))
else:
comp_parent = parent
comp_self = self
corresponding_parent_over_QQ = parent.corresponding_basis_over(QQ)
if (corresponding_parent_over_QQ is None):
comp_parent = parent.realization_of().schur()
comp_self = comp_parent(self)
from sage.combinat.sf.sf import SymmetricFunctions
corresponding_parent_over_QQ = SymmetricFunctions(QQ).schur()
comp_x = comp_parent(x)
result = comp_parent.zero()
for (lam, a) in comp_self.monomial_coefficients().items():
for (mu, b) in comp_x.monomial_coefficients().items():
lam_star_mu = corresponding_parent_over_QQ(lam).itensor(corresponding_parent_over_QQ(mu))
for (nu, c) in lam_star_mu.monomial_coefficients().items():
result += (((a * b) * comp_parent.base_ring()(c)) * comp_parent(nu))
return parent(result)<|docstring|>Return the internal (tensor) product of ``self`` and ``x`` in the
basis of ``self``.
The internal tensor product can be defined as the linear extension
of the definition on power sums
`p_{\lambda} \ast p_{\mu} = \delta_{\lambda,\mu} z_{\lambda}
p_{\lambda}`, where `z_{\lambda} = (1^{r_1} r_1!) (2^{r_2} r_2!)
\cdots` for `\lambda = (1^{r_1} 2^{r_2} \cdots )` and where `\ast`
denotes the internal tensor product.
The internal tensor product is also known as the Kronecker product,
or as the second multiplication on the ring of symmetric functions.
Note that the internal product of any two homogeneous symmetric
functions of equal degrees is a homogeneous symmetric function of the
same degree. On the other hand, the internal product of two homogeneous
symmetric functions of distinct degrees is `0`.
.. NOTE::
The internal product is sometimes referred to as "inner product"
in the literature, but unfortunately this name is shared by a
different operation, namely the Hall inner product
(see :meth:`scalar`).
INPUT:
- ``x`` -- element of the ring of symmetric functions over the
same base ring as ``self``
OUTPUT:
- the internal product of ``self`` with ``x`` (an element of the
ring of symmetric functions in the same basis as ``self``)
The methods :meth:`itensor`, :meth:`internal_product`,
:meth:`kronecker_product`, :meth:`inner_tensor` are all
synonyms.
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([2,1])
sage: b = s([3])
sage: a.itensor(b)
s[2, 1]
sage: c = s([3,2,1])
sage: c.itensor(c)
s[1, 1, 1, 1, 1, 1] + 2*s[2, 1, 1, 1, 1] + 3*s[2, 2, 1, 1] + 2*s[2, 2, 2] + 4*s[3, 1, 1, 1] + 5*s[3, 2, 1] + 2*s[3, 3] + 4*s[4, 1, 1] + 3*s[4, 2] + 2*s[5, 1] + s[6]
There are few quantitative results pertaining to Kronecker products
in general, which makes their computation so difficult. Let us test
a few of them in different bases.
The Kronecker product of any homogeneous symmetric function `f` of
degree `n` with the `n`-th complete homogeneous symmetric function
``h[n]`` (a.k.a. ``s[n]``) is `f`::
sage: h = SymmetricFunctions(ZZ).h()
sage: all( h([5]).itensor(h(p)) == h(p) for p in Partitions(5) )
True
The Kronecker product of a Schur function `s_{\lambda}` with the `n`-th
elementary symmetric function ``e[n]``, where `n = \left| \lambda
\right|`, is `s_{\lambda'}` (where `\lambda'` is the conjugate
partition of `\lambda`)::
sage: F = CyclotomicField(12)
sage: s = SymmetricFunctions(F).s()
sage: e = SymmetricFunctions(F).e()
sage: all( e([5]).itensor(s(p)) == s(p.conjugate()) for p in Partitions(5) )
True
The Kronecker product is commutative::
sage: e = SymmetricFunctions(FiniteField(19)).e()
sage: m = SymmetricFunctions(FiniteField(19)).m()
sage: all( all( e(p).itensor(m(q)) == m(q).itensor(e(p)) for q in Partitions(4) )
....: for p in Partitions(4) )
True
sage: F = FractionField(QQ['q','t'])
sage: mq = SymmetricFunctions(F).macdonald().Q()
sage: mh = SymmetricFunctions(F).macdonald().H()
sage: all( all( mq(p).itensor(mh(r)) == mh(r).itensor(mq(p)) for r in Partitions(4) )
....: for p in Partitions(3) ) # long time
True
Let us check (on examples) Proposition 5.2 of Gelfand, Krob, Lascoux, Leclerc,
Retakh, Thibon, "Noncommutative symmetric functions", :arXiv:`hep-th/9407124`, for
`r = 2`::
sage: e = SymmetricFunctions(FiniteField(29)).e()
sage: s = SymmetricFunctions(FiniteField(29)).s()
sage: m = SymmetricFunctions(FiniteField(29)).m()
sage: def tensor_copr(u, v, w): # computes \mu ((u \otimes v) * \Delta(w)) with
....: # * meaning Kronecker product and \mu meaning the
....: # usual multiplication.
....: result = w.parent().zero()
....: for partition_pair, coeff in w.coproduct().monomial_coefficients().iteritems():
....: result += coeff * w.parent()(u).itensor(partition_pair[0]) * w.parent()(v).itensor(partition_pair[1])
....: return result
sage: all( all( all( tensor_copr(e[u], s[v], m[w]) == (e[u] * s[v]).itensor(m[w])
....: for w in Partitions(5) )
....: for v in Partitions(2) )
....: for u in Partitions(3) ) # long time
True
Some examples from Briand, Orellana, Rosas, "The stability of the Kronecker
products of Schur functions." :arXiv:`0907.4652`::
sage: s = SymmetricFunctions(ZZ).s()
sage: s[2,2].itensor(s[2,2])
s[1, 1, 1, 1] + s[2, 2] + s[4]
sage: s[3,2].itensor(s[3,2])
s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1] + s[5]
sage: s[4,2].itensor(s[4,2])
s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1] + s[6]
An example from p. 220 of Thibon, "Hopf algebras of symmetric functions
and tensor products of symmetric group representations", International
Journal of Algebra and Computation, 0000:0000:0000:0000:0000:0000:0000:0000
sage: s = SymmetricFunctions(QQbar).s()
sage: s[2,1].itensor(s[2,1])
s[1, 1, 1] + s[2, 1] + s[3]
TESTS::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([8,8])
sage: a.itensor(a)
s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2] + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1] + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2] + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2] + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2] + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
sage: s[8].itensor(s[7])
0
sage: s(0).itensor(s(0))
0
sage: s(1).itensor(s(0))
0
sage: s(0).itensor(s(1))
0
sage: s(1).itensor(s(1))
s[]
Same over the ring of integers::
sage: s = SymmetricFunctions(ZZ).s()
sage: a = s([8,8])
sage: a.itensor(a)
s[4, 4, 4, 4] + s[5, 5, 3, 3] + s[5, 5, 5, 1] + s[6, 4, 4, 2] + s[6, 6, 2, 2] + s[6, 6, 4] + s[7, 3, 3, 3] + s[7, 5, 3, 1] + s[7, 7, 1, 1] + s[8, 4, 2, 2] + s[8, 4, 4] + s[8, 6, 2] + s[8, 8] + s[9, 3, 3, 1] + s[9, 5, 1, 1] + s[10, 2, 2, 2] + s[10, 4, 2] + s[10, 6] + s[11, 3, 1, 1] + s[12, 2, 2] + s[12, 4] + s[13, 1, 1, 1] + s[14, 2] + s[16]
sage: s[8].itensor(s[7])
0
sage: s(0).itensor(s(0))
0
sage: s(1).itensor(s(0))
0
sage: s(0).itensor(s(1))
0
sage: s(1).itensor(s(1))
s[]
Theorem 2.1 in Bessenrodt, van Willigenburg, :arXiv:`1105.3170v2`::
sage: s = SymmetricFunctions(ZZ).s()
sage: all( all( max( r[0] for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
....: == sum( min(p[i], q.get_part(i)) for i in range(len(p)) )
....: for p in Partitions(4) )
....: for q in Partitions(4) )
True
sage: all( all( max( len(r) for r in s(p).itensor(s(q)).monomial_coefficients().keys() )
....: == sum( min(p[i], q.conjugate().get_part(i)) for i in range(len(p)) )
....: for p in Partitions(4) )
....: for q in Partitions(4) )
True
Check that the basis and ground ring of ``self`` are preserved::
sage: F = CyclotomicField(12)
sage: s = SymmetricFunctions(F).s()
sage: e = SymmetricFunctions(F).e()
sage: e[3].itensor(s[3])
e[3]
sage: s[3].itensor(e[3])
s[1, 1, 1]
sage: parent(e[3].itensor(s[3]))
Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the elementary basis
sage: parent(s[3].itensor(e[3]))
Symmetric Functions over Cyclotomic Field of order 12 and degree 4 in the Schur basis
.. NOTE::
The currently existing implementation of this function is
technically unsatisfactory. It distinguishes the case when the
base ring is a `\QQ`-algebra (in which case the Kronecker product
can be easily computed using the power sum basis) from the case
where it isn't. In the latter, it does a computation using
universal coefficients, again distinguishing the case when it is
able to compute the "corresponding" basis of the symmetric function
algebra over `\QQ` (using the ``corresponding_basis_over`` hack)
from the case when it isn't (in which case it transforms everything
into the Schur basis, which is slow).<|endoftext|>
|
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