Subset Species

sage.combinat.species.subset_species.SubsetSpecies(*args, **kwds)

Returns the species of subsets.

EXAMPLES:

sage: S = species.SubsetSpecies()
sage: S.generating_series().coefficients(5)
[1, 2, 2, 4/3, 2/3]
sage: S.isotype_generating_series().coefficients(5)
[1, 2, 3, 4, 5]

class sage.combinat.species.subset_species.SubsetSpeciesStructure(parent, labels, list)

Bases: sage.combinat.species.structure.GenericSpeciesStructure

automorphism_group()

Returns the group of permutations whose action on this subset leave it fixed.

EXAMPLES:

sage: F = species.SubsetSpecies()
sage: a = F.structures([1,2,3,4])[6]; a
{1, 3}
sage: a.automorphism_group()
Permutation Group with generators [(2,4), (1,3)]

sage: [a.transport(g) for g in a.automorphism_group()]
[{1, 3}, {1, 3}, {1, 3}, {1, 3}]

canonical_label()

EXAMPLES:

sage: P = species.SubsetSpecies()
sage: S = P.structures(["a", "b", "c"])
sage: [s.canonical_label() for s in S]
[{}, {'a'}, {'a'}, {'a'}, {'a', 'b'}, {'a', 'b'}, {'a', 'b'}, {'a', 'b', 'c'}]

complement()

EXAMPLES:

sage: F = species.SubsetSpecies()
sage: a = F.structures(["a", "b", "c"])[5]; a
{'a', 'c'}
sage: a.complement()
{'b'}

labels()

EXAMPLES:

sage: P = species.SubsetSpecies()
sage: S = P.structures(["a", "b", "c"])
sage: [s.labels() for s in S]
[[], ['a'], ['b'], ['c'], ['a', 'b'], ['a', 'c'], ['b', 'c'], ['a', 'b', 'c']]

transport(perm)

Returns the transport of this subset along the permutation perm.

EXAMPLES:

sage: F = species.SubsetSpecies()
sage: a = F.structures(["a", "b", "c"])[5]; a
{'a', 'c'}
sage: p = PermutationGroupElement((1,2))
sage: a.transport(p)
{'b', 'c'}
sage: p = PermutationGroupElement((1,3))
sage: a.transport(p)
{'a', 'c'}

class sage.combinat.species.subset_species.SubsetSpecies_class(min=None, max=None, weight=None)

Bases: sage.combinat.species.species.GenericCombinatorialSpecies