def expand_as_sf(self, n, alphabet='x'):
"""
Returns the expansion of a cycle index series as a symmetric function in
``n`` variables.
Specifically, this returns a :class:`~sage.combinat.species.series.LazyPowerSeries` whose
ith term is obtained by calling :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.expand`
on the ith term of ``self``.
This relies on the (standard) interpretation of a cycle index series as a symmetric function
in the power sum basis.
INPUT:
- ``self`` -- a cycle index series
- ``n`` -- a positive integer
- ``alphabet`` -- a variable for the expansion (default: `x`)
EXAMPLES::
sage: from sage.combinat.species.set_species import SetSpecies
sage: SetSpecies().cycle_index_series().expand_as_sf(2).coefficients(4)
[1, x0 + x1, x0^2 + x0*x1 + x1^2, x0^3 + x0^2*x1 + x0*x1^2 + x1^3]
"""
expanded_poly_ring = self.coefficient(0).expand(n, alphabet).parent()
LPSR = LazyPowerSeriesRing(expanded_poly_ring)
expander_gen = (LPSR.term(self.coefficient(i).expand(n, alphabet), i) for i in _integers_from(0))
return LPSR.sum_generator(expander_gen)
def expand_as_sf(self, n, alphabet='x'):
"""
Returns the expansion of a cycle index series as a symmetric function in
``n`` variables.
Specifically, this returns a :class:`~sage.combinat.species.series.LazyPowerSeries` whose
ith term is obtained by calling :meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.expand`
on the ith term of ``self``.
This relies on the (standard) interpretation of a cycle index series as a symmetric function
in the power sum basis.
INPUT:
- ``self`` -- a cycle index series
- ``n`` -- a positive integer
- ``alphabet`` -- a variable for the expansion (default: `x`)
EXAMPLES::
sage: from sage.combinat.species.set_species import SetSpecies
sage: SetSpecies().cycle_index_series().expand_as_sf(2).coefficients(4)
[1, x0 + x1, x0^2 + x0*x1 + x1^2, x0^3 + x0^2*x1 + x0*x1^2 + x1^3]
"""
expanded_poly_ring = self.coefficient(0).expand(n, alphabet).parent()
LPSR = LazyPowerSeriesRing(expanded_poly_ring)
expander_gen = (LPSR.term(self.coefficient(i).expand(n, alphabet), i)
for i in _integers_from(0))
return LPSR.sum_generator(expander_gen)
def __init__(self, R):
"""
EXAMPLES::
sage: from sage.combinat.species.generating_series import ExponentialGeneratingSeriesRing
sage: R = ExponentialGeneratingSeriesRing(QQ)
sage: R == loads(dumps(R))
True
"""
LazyPowerSeriesRing.__init__(self, R, ExponentialGeneratingSeries)
def __init__(self, R):
"""
EXAMPLES::
sage: from sage.combinat.species.generating_series import ExponentialGeneratingSeriesRing
sage: R = ExponentialGeneratingSeriesRing(QQ)
sage: R == loads(dumps(R))
True
"""
LazyPowerSeriesRing.__init__(self, R, ExponentialGeneratingSeries)
def __init__(self, R):
"""
EXAMPLES::
sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: R = CycleIndexSeriesRing(QQ); R
Cycle Index Series Ring over Symmetric Functions over Rational Field in the powersum basis
sage: R == loads(dumps(R))
True
"""
R = SymmetricFunctions(R).power()
LazyPowerSeriesRing.__init__(self, R, CycleIndexSeries)
def __init__(self, R):
"""
EXAMPLES::
sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: R = CycleIndexSeriesRing(QQ); R
Cycle Index Series Ring over Symmetric Functions over Rational Field in the powersum basis
sage: R == loads(dumps(R))
True
"""
R = SymmetricFunctions(R).power()
LazyPowerSeriesRing.__init__(self, R, CycleIndexSeries)
