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)
__repr__()

EXAMPLES:

sage: S = species.SubsetSpecies()
sage: a = S.structures(["a","b","c"]).random_element(); a
{}
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)
__init__(min=None, max=None, weight=None)

EXAMPLES:

sage: S = species.SubsetSpecies()
sage: c = S.generating_series().coefficients(3)
sage: S._check()
True
sage: S == loads(dumps(S))
True
static _cached_constructor(*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]
_cis(series_ring, base_ring)

The cycle index series for the species of subsets is given by

exp \left( 2 \cdot \sum_{n=1}^\infty \frac{x_n}{n} \right).

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]]
_cis_gen(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]]
_default_structure_class
alias of SubsetSpeciesStructure
_gs_iterator(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]
_isotypes(structure_class, labels)

EXAMPLES:

sage: S = species.SubsetSpecies()
sage: S.isotypes([1,2]).list()
[{}, {1}, {1, 2}]
sage: S.isotypes(['a','b']).list()
[{}, {'a'}, {'a', 'b'}]
_itgs_iterator(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]
_structures(structure_class, labels)

EXAMPLES:

sage: S = species.SubsetSpecies()
sage: S.structures([1,2]).list()
[{}, {1}, {2}, {1, 2}]
sage: S.structures(['a','b']).list()
[{}, {'a'}, {'b'}, {'a', 'b'}]

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