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 _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)
res = ciset.composition(ciset - 1)
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)
res = ciset.composition(ciset - 1)
if self.is_weighted():
res *= self._weight
return res
def SimpleGraphSpecies():
"""
Returns the species of simple graphs.
EXAMPLES::
sage: S = species.SimpleGraphSpecies()
sage: S.generating_series().counts(10)
[1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456, 68719476736]
sage: S.cycle_index_series().coefficients(5)
[p[],
p[1],
p[1, 1] + p[2],
4/3*p[1, 1, 1] + 2*p[2, 1] + 2/3*p[3],
8/3*p[1, 1, 1, 1] + 4*p[2, 1, 1] + 2*p[2, 2] + 4/3*p[3, 1] + p[4]]
sage: S.isotype_generating_series().coefficients(6)
[1, 1, 2, 4, 11, 34]
TESTS::
sage: seq = S.isotype_generating_series().counts(6)[1:]
sage: oeis(seq)[0] # optional -- internet
A000088: Number of graphs on n unlabeled nodes.
::
sage: seq = S.generating_series().counts(10)[1:]
sage: oeis(seq)[0] # optional -- internet
A006125: a(n) = 2^(n(n-1)/2).
"""
E = SetSpecies()
E2 = SetSpecies(size=2)
WP = SubsetSpecies()
P2 = E2 * E
return WP.functorial_composition(P2)
def _cis_term(self, base_ring):
"""
EXAMPLES::
sage: F = species.CharacteristicSpecies(2)
sage: g = F.cycle_index_series()
sage: g.coefficients(5)
[0, 0, 1/2*p[1, 1] + 1/2*p[2], 0, 0]
"""
cis = SetSpecies(weight=self._weight).cycle_index_series(base_ring)
return cis.coefficient(self._n)
def BinaryForestSpecies():
"""
Returns the species of binary forests. Binary forests are defined
as sets of binary trees.
EXAMPLES::
sage: F = species.BinaryForestSpecies()
sage: F.generating_series().counts(10)
[1, 1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353]
sage: F.isotype_generating_series().counts(10)
[1, 1, 2, 4, 10, 26, 77, 235, 758, 2504]
sage: F.cycle_index_series().coefficients(7)
[p[],
p[1],
3/2*p[1, 1] + 1/2*p[2],
19/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3],
193/24*p[1, 1, 1, 1] + 3/4*p[2, 1, 1] + 5/8*p[2, 2] + 1/3*p[3, 1] + 1/4*p[4],
907/40*p[1, 1, 1, 1, 1] + 19/12*p[2, 1, 1, 1] + 5/8*p[2, 2, 1] + 1/2*p[3, 1, 1] + 1/6*p[3, 2] + 1/4*p[4, 1] + 1/5*p[5],
49171/720*p[1, 1, 1, 1, 1, 1] + 193/48*p[2, 1, 1, 1, 1] + 15/16*p[2, 2, 1, 1] + 61/48*p[2, 2, 2] + 19/18*p[3, 1, 1, 1] + 1/6*p[3, 2, 1] + 7/18*p[3, 3] + 3/8*p[4, 1, 1] + 1/8*p[4, 2] + 1/5*p[5, 1] + 1/6*p[6]]
TESTS::
sage: seq = F.isotype_generating_series().counts(10)[1:] #optional
sage: number, name, sseq = sloane_find(seq)[0] #optional
sage: print number, name #optional
52854 Number of unordered forests on n nodes.
"""
B = BinaryTreeSpecies()
S = SetSpecies()
F = S(B)
return F
def _cis(self, series_ring, base_ring):
r"""
The cycle index series for the species of subsets satisfies
.. math::
Z_{\mathfrak{p}} = Z_{\mathcal{E}} \cdot Z_{\mathcal{E}}
EXAMPLES::
sage: S = species.SubsetSpecies()
sage: S.cycle_index_series().coefficients(5)
[p[],
2*p[1],
2*p[1, 1] + p[2],
4/3*p[1, 1, 1] + 2*p[2, 1] + 2/3*p[3],
2/3*p[1, 1, 1, 1] + 2*p[2, 1, 1] + 1/2*p[2, 2] + 4/3*p[3, 1] + 1/2*p[4]]
"""
ciset = SetSpecies().cycle_index_series(base_ring)
res = ciset**2
if self.is_weighted():
res *= self._weight
return res