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]
EXAMPLES:
sage: S = species.SubsetSpecies()
sage: a = S.structures(["a","b","c"]).random_element(); a
{}
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}]
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'}]
EXAMPLES:
sage: F = species.SubsetSpecies()
sage: a = F.structures(["a", "b", "c"])[5]; a
{'a', 'c'}
sage: a.complement()
{'b'}
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']]
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'}
EXAMPLES:
sage: S = species.SubsetSpecies()
sage: c = S.generating_series().coefficients(3)
sage: S._check()
True
sage: S == loads(dumps(S))
True
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]
The cycle index series for the species of subsets is given by
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]]
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]]
The generating series for the species of subsets is
.
EXAMPLES:
sage: S = species.SubsetSpecies()
sage: S.generating_series().coefficients(5)
[1, 2, 2, 4/3, 2/3]
EXAMPLES:
sage: S = species.SubsetSpecies()
sage: S.isotypes([1,2]).list()
[{}, {1}, {1, 2}]
sage: S.isotypes(['a','b']).list()
[{}, {'a'}, {'a', 'b'}]
The generating series for the species of subsets is
.
EXAMPLES:
sage: S = species.SubsetSpecies()
sage: S.isotype_generating_series().coefficients(5)
[1, 2, 3, 4, 5]
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'}]