Returns the species of sets.
EXAMPLES:
sage: E = species.SetSpecies()
sage: E.structures([1,2,3]).list()
[{1, 2, 3}]
sage: E.isotype_generating_series().coefficients(4)
[1, 1, 1, 1]
EXAMPLES:
sage: S = species.SetSpecies()
sage: a = S.structures(["a","b","c"]).random_element(); a
{'a', 'b', 'c'}
Returns the group of permutations whose action on this set leave it fixed. For the species of sets, there is only one isomorphism class, so every permutation is in its automorphism group.
EXAMPLES:
sage: F = species.SetSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
{'a', 'b', 'c'}
sage: a.automorphism_group()
Symmetric group of order 3! as a permutation group
EXAMPLES:
sage: S = species.SetSpecies()
sage: a = S.structures(["a","b","c"]).random_element(); a
{'a', 'b', 'c'}
sage: a.canonical_label()
{'a', 'b', 'c'}
Returns the transport of this set along the permutation perm.
EXAMPLES:
sage: F = species.SetSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
{'a', 'b', 'c'}
sage: p = PermutationGroupElement((1,2))
sage: a.transport(p)
{'a', 'b', 'c'}
EXAMPLES:
sage: S = species.SetSpecies()
sage: c = S.generating_series().coefficients(3)
sage: S._check()
True
sage: S == loads(dumps(S))
True
Returns the species of sets.
EXAMPLES:
sage: E = species.SetSpecies()
sage: E.structures([1,2,3]).list()
[{1, 2, 3}]
sage: E.isotype_generating_series().coefficients(4)
[1, 1, 1, 1]
The cycle index series for the species of sets is given by
.EXAMPLES:
sage: S = species.SetSpecies()
sage: g = S.cycle_index_series()
sage: g.coefficients(5)
[p[],
p[1],
1/2*p[1, 1] + 1/2*p[2],
1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3],
1/24*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] + 1/8*p[2, 2] + 1/3*p[3, 1] + 1/4*p[4]]
EXAMPLES:
sage: S = species.SetSpecies()
sage: g = S._cis_gen(QQ)
sage: [g.next() for i in range(5)]
[0, p[1], 1/2*p[2], 1/3*p[3], 1/4*p[4]]
The generating series for the species of sets is given by
.
EXAMPLES:
sage: S = species.SetSpecies()
sage: g = S.generating_series()
sage: g.coefficients(10)
[1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880]
sage: [g.count(i) for i in range(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
EXAMPLES:
sage: S = species.SetSpecies()
sage: S.structures([1,2,3]).list()
[{1, 2, 3}]
The isomorphism type generating series for the species of sets is
.
EXAMPLES:
sage: S = species.SetSpecies()
sage: g = S.isotype_generating_series()
sage: g.coefficients(10)
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: [g.count(i) for i in range(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
EXAMPLES:
sage: S = species.SetSpecies()
sage: S.structures([1,2,3]).list()
[{1, 2, 3}]