EXAMPLES:
sage: G = species.SimpleGraphSpecies()
sage: c = G.generating_series().coefficients(2)
sage: type(G)
<class 'sage.combinat.species.functorial_composition_species.FunctorialCompositionSpecies_class'>
sage: G == loads(dumps(G))
True
sage: G._check() #False due to isomorphism types not being implemented
False
EXAMPLES:
sage: G = species.SimpleGraphSpecies()
sage: G.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]]
EXAMPLES:
sage: G = species.SimpleGraphSpecies()
sage: G.generating_series().coefficients(5)
[1, 1, 1, 4/3, 8/3]
There is no known algorithm for efficiently generating the isomorphism types of the functorial composition of two species.
EXAMPLES:
sage: G = species.SimpleGraphSpecies()
sage: G.isotypes([1,2,3]).list()
...
NotImplementedError
EXAMPLES:
sage: G = species.SimpleGraphSpecies()
sage: G.isotype_generating_series().coefficients(5)
[1, 1, 2, 4, 11]
EXAMPLES:
sage: G = species.SimpleGraphSpecies()
sage: G.structures([1,2,3]).list()
[{},
{{1, 2}*{3}},
{{1, 3}*{2}},
{{2, 3}*{1}},
{{1, 2}*{3}, {1, 3}*{2}},
{{1, 2}*{3}, {2, 3}*{1}},
{{1, 3}*{2}, {2, 3}*{1}},
{{1, 2}*{3}, {1, 3}*{2}, {2, 3}*{1}}]
Returns the weight ring for this species. This is determined by asking Sage’s coercion model what the result is when you multiply (and add) elements of the weight rings for each of the operands.
EXAMPLES:
sage: G = species.SimpleGraphSpecies()
sage: G.weight_ring()
Rational Field