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 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 _times_gen(self, structure_class, attr, labels):
"""
EXAMPLES::
sage: S = species.SetSpecies()
sage: F = S * S
sage: list(F._times_gen(F._default_structure_class, 'structures',[1,2]))
[{}*{1, 2}, {1}*{2}, {2}*{1}, {1, 2}*{}]
"""
c = lambda F, n: F.generating_series().coefficient(n)
S = SubsetSpecies()
for u in getattr(S, attr)(labels):
vl = u.complement().labels()
ul = u.labels()
if c(self._F, len(ul)) == 0 or c(self._G, len(vl)) == 0:
continue
for x in getattr(self._F, attr)(ul):
for y in getattr(self._G, attr)(vl):
yield structure_class(self, labels, u, x, y)