def _cis(self, series_ring, base_ring):
r"""
The cycle index series for the species of partitions is given by
.. math::
exp \sum_{n \ge 1} \frac{1}{n} \left( exp \left( \sum_{k \ge 1} \frac{x_{kn}}{k} \right) -1 \right).
EXAMPLES::
sage: P = species.PartitionSpecies()
sage: g = P.cycle_index_series()
sage: g.coefficients(5)
[p[],
p[1],
p[1, 1] + p[2],
5/6*p[1, 1, 1] + 3/2*p[2, 1] + 2/3*p[3],
5/8*p[1, 1, 1, 1] + 7/4*p[2, 1, 1] + 7/8*p[2, 2] + p[3, 1] + 3/4*p[4]]
"""
ciset = SetSpecies().cycle_index_series(base_ring)
CIS = ciset.parent()
res = CIS.sum_generator(((1/n)*ciset).stretch(n) for n in _integers_from(ZZ(1))).exponential()
if self.is_weighted():
res *= self._weight
return res
def _cis(self, series_ring, base_ring):
r"""
The cycle index series for the species of partitions is given by
.. math::
exp \sum_{n \ge 1} \frac{1}{n} \left( exp \left( \sum_{k \ge 1} \frac{x_{kn}}{k} \right) -1 \right).
EXAMPLES::
sage: P = species.PartitionSpecies()
sage: g = P.cycle_index_series()
sage: g.coefficients(5)
[p[],
p[1],
p[1, 1] + p[2],
5/6*p[1, 1, 1] + 3/2*p[2, 1] + 2/3*p[3],
5/8*p[1, 1, 1, 1] + 7/4*p[2, 1, 1] + 7/8*p[2, 2] + p[3, 1] + 3/4*p[4]]
"""
ciset = SetSpecies().cycle_index_series(base_ring)
CIS = ciset.parent()
res = CIS.sum_generator(((1 / n) * ciset).stretch(n)
for n in _integers_from(ZZ(1))).exponential()
if self.is_weighted():
res *= self._weight
return res
