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
def _gs_iterator(self, base_ring):
r"""
EXAMPLES::
sage: P = species.PartitionSpecies()
sage: g = P.generating_series()
sage: g.coefficients(5)
[1, 1, 1, 5/6, 5/8]
"""
from sage.combinat.combinat import bell_number
for n in _integers_from(0):
yield self._weight*base_ring(bell_number(n)/factorial_stream[n])
def _gs_iterator(self, base_ring):
r"""
EXAMPLES::
sage: P = species.PartitionSpecies()
sage: g = P.generating_series()
sage: g.coefficients(5)
[1, 1, 1, 5/6, 5/8]
"""
from sage.combinat.combinat import bell_number
for n in _integers_from(0):
yield self._weight*base_ring(bell_number(n)/factorial_stream[n])
def _itgs_iterator(self, base_ring):
"""
The generating series for the species of subsets is
`e^{2x}`.
EXAMPLES::
sage: S = species.SubsetSpecies()
sage: S.isotype_generating_series().coefficients(5)
[1, 2, 3, 4, 5]
"""
for n in _integers_from(1):
yield base_ring(n)
def _cis_iterator(self, base_ring):
"""
EXAMPLES::
sage: L = species.LinearOrderSpecies()
sage: g = L.cycle_index_series()
sage: g.coefficients(5)
[p[], p[1], p[1, 1], p[1, 1, 1], p[1, 1, 1, 1]]
"""
from sage.combinat.sf.sf import SymmetricFunctions
p = SymmetricFunctions(base_ring).power()
for n in _integers_from(0):
yield p([1]*n)
def _gs_iterator(self, base_ring):
"""
The generating series for the species of subsets is
`e^{2x}`.
EXAMPLES::
sage: S = species.SubsetSpecies()
sage: S.generating_series().coefficients(5)
[1, 2, 2, 4/3, 2/3]
"""
for n in _integers_from(0):
yield base_ring(2) ** n / base_ring(factorial_stream[n])
def _itgs_iterator(self, base_ring):
"""
The generating series for the species of subsets is
`e^{2x}`.
EXAMPLES::
sage: S = species.SubsetSpecies()
sage: S.isotype_generating_series().coefficients(5)
[1, 2, 3, 4, 5]
"""
for n in _integers_from(1):
yield base_ring(n)
def _gs_iterator(self, base_ring):
"""
The generating series for the species of subsets is
`e^{2x}`.
EXAMPLES::
sage: S = species.SubsetSpecies()
sage: S.generating_series().coefficients(5)
[1, 2, 2, 4/3, 2/3]
"""
for n in _integers_from(0):
yield base_ring(2)**n / base_ring(factorial_stream[n])
def _cis_gen(self, base_ring):
"""
EXAMPLES::
sage: S = species.SetSpecies()
sage: g = S._cis_gen(QQ)
sage: [next(g) for i in range(5)]
[0, p[1], 1/2*p[2], 1/3*p[3], 1/4*p[4]]
"""
from sage.combinat.sf.sf import SymmetricFunctions
p = SymmetricFunctions(base_ring).power()
yield p(0)
for n in _integers_from(1):
yield p([n])/n
def _cis_gen(self, base_ring):
"""
EXAMPLES::
sage: S = species.SubsetSpecies()
sage: g = S._cis_gen(QQ)
sage: [g.next() for i in range(5)]
[0, 2*p[1], p[2], 2/3*p[3], 1/2*p[4]]
"""
from sage.combinat.sf.all import SFAPower
p = SFAPower(base_ring)
yield base_ring(0)
for n in _integers_from(ZZ(1)):
yield 2*p([n])/n
def _cis_gen(self, base_ring):
"""
EXAMPLES::
sage: S = species.SubsetSpecies()
sage: g = S._cis_gen(QQ)
sage: [g.next() for i in range(5)]
[0, 2*p[1], p[2], 2/3*p[3], 1/2*p[4]]
"""
from sage.combinat.sf.all import SFAPower
p = SFAPower(base_ring)
yield base_ring(0)
for n in _integers_from(ZZ(1)):
yield 2 * p([n]) / n
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.partitions import number_of_partitions
for n in _integers_from(0):
yield base_ring(number_of_partitions(n))
def _gs_iterator(self, base_ring):
r"""
The generating series for the species of sets is given by
`e^x`.
EXAMPLES::
sage: S = species.SetSpecies()
sage: g = S.generating_series()
sage: g.coefficients(10)
[1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880]
sage: [g.count(i) for i in range(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
"""
for n in _integers_from(0):
yield base_ring(self._weight/factorial_stream[n])
def _cis_iterator(self, base_ring):
r"""
The cycle index series of the species of cyclic permutations is
given by
.. math::
-\sum_{k=1}^\infty \phi(k)/k * log(1 - x_k)
which is equal to
.. math::
\sum_{n=1}^\infty \frac{1}{n} * \sum_{k|n} \phi(k) * x_k^{n/k}
.
EXAMPLES::
sage: P = species.CycleSpecies()
sage: cis = P.cycle_index_series()
sage: cis.coefficients(7)
[0,
p[1],
1/2*p[1, 1] + 1/2*p[2],
1/3*p[1, 1, 1] + 2/3*p[3],
1/4*p[1, 1, 1, 1] + 1/4*p[2, 2] + 1/2*p[4],
1/5*p[1, 1, 1, 1, 1] + 4/5*p[5],
1/6*p[1, 1, 1, 1, 1, 1] + 1/6*p[2, 2, 2] + 1/3*p[3, 3] + 1/3*p[6]]
"""
from sage.combinat.sf.sf import SymmetricFunctions
p = SymmetricFunctions(base_ring).power()
zero = base_ring(0)
yield zero
for n in _integers_from(1):
res = zero
for k in divisors(n):
res += euler_phi(k) * p([k]) ** (n // k)
res /= n
yield self._weight * res
def _cis_iterator(self, base_ring):
r"""
The cycle index series of the species of cyclic permutations is
given by
.. math::
-\sum_{k=1}^\infty \phi(k)/k * log(1 - x_k)
which is equal to
.. math::
\sum_{n=1}^\infty \frac{1}{n} * \sum_{k|n} \phi(k) * x_k^{n/k}
.
EXAMPLES::
sage: P = species.CycleSpecies()
sage: cis = P.cycle_index_series()
sage: cis.coefficients(7)
[0,
p[1],
1/2*p[1, 1] + 1/2*p[2],
1/3*p[1, 1, 1] + 2/3*p[3],
1/4*p[1, 1, 1, 1] + 1/4*p[2, 2] + 1/2*p[4],
1/5*p[1, 1, 1, 1, 1] + 4/5*p[5],
1/6*p[1, 1, 1, 1, 1, 1] + 1/6*p[2, 2, 2] + 1/3*p[3, 3] + 1/3*p[6]]
"""
from sage.combinat.sf.all import SFAPower
p = SFAPower(base_ring)
zero = base_ring(0)
yield zero
for n in _integers_from(1):
res = zero
for k in divisors(n):
res += euler_phi(k)*p([k])**(n//k)
res /= n
yield self._weight*res
def _cis_gen(self, base_ring, n):
"""
EXAMPLES::
sage: P = species.PermutationSpecies()
sage: g = P._cis_gen(QQ, 2)
sage: [g.next() for i in range(10)]
[p[], 0, p[2], 0, p[2, 2], 0, p[2, 2, 2], 0, p[2, 2, 2, 2], 0]
"""
from sage.combinat.sf.sf import SymmetricFunctions
p = SymmetricFunctions(base_ring).power()
pn = p([n])
n = n - 1
yield p(1)
for k in _integers_from(1):
for i in range(n):
yield base_ring(0)
yield pn**k
def _cis(self, series_ring, base_ring):
r"""
The cycle index series for the species of permutations is given by
.. math::
\prod{n=1}^\infty \frac{1}{1-x_n}.
EXAMPLES::
sage: P = species.PermutationSpecies()
sage: g = P.cycle_index_series()
sage: g.coefficients(5)
[p[],
p[1],
p[1, 1] + p[2],
p[1, 1, 1] + p[2, 1] + p[3],
p[1, 1, 1, 1] + p[2, 1, 1] + p[2, 2] + p[3, 1] + p[4]]
"""
CIS = series_ring
return CIS.product_generator( CIS(self._cis_gen(base_ring, i)) for i in _integers_from(ZZ(1)) )
def _gs_iterator(self, base_ring):
r"""
The generating series for cyclic permutations is
`-\log(1-x) = \sum_{n=1}^\infty x^n/n`.
EXAMPLES::
sage: P = species.CycleSpecies()
sage: g = P.generating_series()
sage: g.coefficients(10)
[0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9]
TESTS::
sage: P = species.CycleSpecies()
sage: g = P.generating_series(RR)
sage: g.coefficients(3)
[0.000000000000000, 1.00000000000000, 0.500000000000000]
"""
one = base_ring(1)
yield base_ring(0)
for n in _integers_from(ZZ(1)):
yield self._weight * one / n
def _gs_iterator(self, base_ring):
r"""
The generating series for cyclic permutations is
`-\log(1-x) = \sum_{n=1}^\infty x^n/n`.
EXAMPLES::
sage: P = species.CycleSpecies()
sage: g = P.generating_series()
sage: g.coefficients(10)
[0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9]
TESTS::
sage: P = species.CycleSpecies()
sage: g = P.generating_series(RR)
sage: g.coefficients(3)
[0.000000000000000, 1.00000000000000, 0.500000000000000]
"""
one = base_ring(1)
yield base_ring(0)
for n in _integers_from(ZZ(1)):
yield self._weight * one / n
