Cycle Species

sage.combinat.species.cycle_species.CycleSpecies(*args, **kwds)

Returns the species of cycles.

EXAMPLES:

sage: C = species.CycleSpecies(); C
Cyclic permutation species
sage: C.structures([1,2,3,4]).list()
[(1, 2, 3, 4),
 (1, 2, 4, 3),
 (1, 3, 2, 4),
 (1, 3, 4, 2),
 (1, 4, 2, 3),
 (1, 4, 3, 2)]

TESTS: We check to verify that the caching of species is actually working.

sage: species.CycleSpecies() is species.CycleSpecies()
True

class sage.combinat.species.cycle_species.CycleSpeciesStructure(parent, labels, list)

Bases: sage.combinat.species.structure.GenericSpeciesStructure

automorphism_group()

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

EXAMPLES:

sage: P = species.CycleSpecies()
sage: a = P.structures([1, 2, 3, 4]).random_element(); a
(1, 2, 3, 4)
sage: a.automorphism_group()
Permutation Group with generators [(1,2,3,4)]

sage: [a.transport(perm) for perm in a.automorphism_group()]
[(1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4)]

canonical_label()

EXAMPLES:

sage: P = species.CycleSpecies()
sage: P.structures(["a","b","c"]).random_element().canonical_label()
('a', 'b', 'c')

permutation_group_element()

Returns this cycle as a permutation group element.

EXAMPLES:

sage: F = species.CycleSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
('a', 'b', 'c')
sage: a.permutation_group_element()
(1,2,3)

transport(perm)

Returns the transport of this structure along the permutation perm.

EXAMPLES:

sage: F = species.CycleSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
('a', 'b', 'c')
sage: p = PermutationGroupElement((1,2))
sage: a.transport(p)
('a', 'c', 'b')

class sage.combinat.species.cycle_species.CycleSpecies_class(min=None, max=None, weight=None)

Bases: sage.combinat.species.species.GenericCombinatorialSpecies