def _wronskian_invdeterminant(self, weight_parity = 0) :
r"""
The inverse determinant of `W`, which in the considered cases is always a negative
power of the eta function. See the thesis of Nils Skoruppa.
INPUT:
- ``weight_parity`` -- An integer (default: `0`).
"""
try :
if weight_parity % 2 == 0 :
wronskian_invdeterminant = self._wronskian_invdeterminant_even
else :
wronskian_invdeterminant = self._wronskian_invdeterminant_odd
except AttributeError :
m = self.jacobi_index()
if weight_parity % 2 == 0 :
pw = (m + 1) * (2 * m + 1)
else :
pw = (m - 1) * (2 * m - 1)
qexp_prec = self._qexp_precision()
wronskian_invdeterminant = self.integral_power_series_ring() \
( [ number_of_partitions(n) for n in xrange(qexp_prec) ] ) \
.add_bigoh(qexp_prec) ** pw
if weight_parity % 2 == 0 :
self._wronskian_invdeterminant_even = wronskian_invdeterminant
else :
self._wronskian_invdeterminant_odd = wronskian_invdeterminant
return wronskian_invdeterminant
def _wronskian_invdeterminant(self, weight_parity=0):
r"""
The inverse determinant of `W`, which in the considered cases is always a negative
power of the eta function. See the thesis of Nils Skoruppa.
INPUT:
- ``weight_parity`` -- An integer (default: `0`).
"""
try:
if weight_parity % 2 == 0:
wronskian_invdeterminant = self._wronskian_invdeterminant_even
else:
wronskian_invdeterminant = self._wronskian_invdeterminant_odd
except AttributeError:
m = self.jacobi_index()
if weight_parity % 2 == 0:
pw = (m + 1) * (2 * m + 1)
else:
pw = (m - 1) * (2 * m - 1)
qexp_prec = self._qexp_precision()
wronskian_invdeterminant = self.integral_power_series_ring() \
( [ number_of_partitions(n) for n in xrange(qexp_prec) ] ) \
.add_bigoh(qexp_prec) ** pw
if weight_parity % 2 == 0:
self._wronskian_invdeterminant_even = wronskian_invdeterminant
else:
self._wronskian_invdeterminant_odd = wronskian_invdeterminant
return wronskian_invdeterminant
def _eta_power(self):
try:
return self.__eta_power
except AttributeError:
qexp_prec = self._get_maass_form_qexp_prec()
self.__eta_power = self.integral_power_series_ring() \
([number_of_partitions(n) for n in xrange(qexp_prec)], qexp_prec)**6
return self.__eta_power
def _eta_power(self) :
try :
return self.__eta_power
except AttributeError :
qexp_prec = self._get_maass_form_qexp_prec()
self.__eta_power = self.integral_power_series_ring() \
([number_of_partitions(n) for n in xrange(qexp_prec)], qexp_prec)**6
return self.__eta_power
def _itgs_iterator(self, base_ring):
r"""
The isomorphism type generating series is given by
`\frac{1}{1-x}`.
EXAMPLES::
sage: P = species.PermutationSpecies()
sage: g = P.isotype_generating_series()
sage: g.coefficients(10)
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]
"""
from sage.combinat.partition import number_of_partitions
for n in _integers_from(0):
yield base_ring(number_of_partitions(n))
def _itgs_iterator(self, base_ring):
r"""
The isomorphism type generating series is given by
`\frac{1}{1-x}`.
EXAMPLES::
sage: P = species.PartitionSpecies()
sage: g = P.isotype_generating_series()
sage: g.coefficients(10)
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]
"""
from sage.combinat.partition import number_of_partitions
for n in _integers_from(0):
yield self._weight*base_ring(number_of_partitions(n))
def _eta_factor(self):
r"""
The inverse determinant of `W`, which in these cases is always a negative
power of the eta function.
"""
try:
return self.__eta_factor
except AttributeError:
m = self.__precision.jacobi_index()
pw = (m + 1) * (2 * m + 1)
qexp_prec = self._qexp_precision()
self.__eta_factor = self.integral_power_series_ring() \
( [ number_of_partitions(n) for n in xrange(qexp_prec) ] ) \
.add_bigoh(qexp_prec) ** pw
return self.__eta_factor
def _eta_factor(self) :
r"""
The inverse determinant of `W`, which in these cases is always a negative
power of the eta function.
"""
try :
return self.__eta_factor
except AttributeError :
m = self.__precision.jacobi_index()
pw = (m + 1) * (2 * m + 1)
qexp_prec = self._qexp_precision()
self.__eta_factor = self.integral_power_series_ring() \
( [ number_of_partitions(n) for n in xrange(qexp_prec) ] ) \
.add_bigoh(qexp_prec) ** pw
return self.__eta_factor
