Cycle Species¶
- class sage.combinat.species.cycle_species.CycleSpecies(min=None, max=None, weight=None)[source]¶
- Bases: - GenericCombinatorialSpecies,- UniqueRepresentation- Return 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)] - >>> from sage.all import * >>> C = species.CycleSpecies(); C Cyclic permutation species >>> C.structures([Integer(1),Integer(2),Integer(3),Integer(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)] 
- class sage.combinat.species.cycle_species.CycleSpeciesStructure(parent, labels, list)[source]¶
- Bases: - GenericSpeciesStructure- automorphism_group()[source]¶
- Return 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])[0]; a (1, 2, 3, 4) sage: a.automorphism_group() # needs sage.groups Permutation Group with generators [(1,2,3,4)] - >>> from sage.all import * >>> P = species.CycleSpecies() >>> a = P.structures([Integer(1), Integer(2), Integer(3), Integer(4)])[Integer(0)]; a (1, 2, 3, 4) >>> a.automorphism_group() # needs sage.groups Permutation Group with generators [(1,2,3,4)] - sage: [a.transport(perm) for perm in a.automorphism_group()] # needs sage.groups [(1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4)] - >>> from sage.all import * >>> [a.transport(perm) for perm in a.automorphism_group()] # needs sage.groups [(1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4)] 
 - canonical_label()[source]¶
- EXAMPLES: - sage: P = species.CycleSpecies() sage: P.structures(["a","b","c"]).random_element().canonical_label() ('a', 'b', 'c') - >>> from sage.all import * >>> P = species.CycleSpecies() >>> P.structures(["a","b","c"]).random_element().canonical_label() ('a', 'b', 'c') 
 - permutation_group_element()[source]¶
- Return this cycle as a permutation group element. - EXAMPLES: - sage: F = species.CycleSpecies() sage: a = F.structures(["a", "b", "c"])[0]; a ('a', 'b', 'c') sage: a.permutation_group_element() # needs sage.groups (1,2,3) - >>> from sage.all import * >>> F = species.CycleSpecies() >>> a = F.structures(["a", "b", "c"])[Integer(0)]; a ('a', 'b', 'c') >>> a.permutation_group_element() # needs sage.groups (1,2,3) 
 - transport(perm)[source]¶
- Return the transport of this structure along the permutation perm. - EXAMPLES: - sage: F = species.CycleSpecies() sage: a = F.structures(["a", "b", "c"])[0]; a ('a', 'b', 'c') sage: p = PermutationGroupElement((1,2)) # needs sage.groups sage: a.transport(p) # needs sage.groups ('a', 'c', 'b') - >>> from sage.all import * >>> F = species.CycleSpecies() >>> a = F.structures(["a", "b", "c"])[Integer(0)]; a ('a', 'b', 'c') >>> p = PermutationGroupElement((Integer(1),Integer(2))) # needs sage.groups >>> a.transport(p) # needs sage.groups ('a', 'c', 'b') 
 
- sage.combinat.species.cycle_species.CycleSpecies_class[source]¶
- alias of - CycleSpecies