Permutation species¶
- class sage.combinat.species.permutation_species.PermutationSpecies(min=None, max=None, weight=None)[source]¶
- Bases: - GenericCombinatorialSpecies,- UniqueRepresentation- Return the species of permutations. - EXAMPLES: - sage: P = species.PermutationSpecies() sage: P.generating_series()[0:5] [1, 1, 1, 1, 1] sage: P.isotype_generating_series()[0:5] [1, 1, 2, 3, 5] sage: P = species.PermutationSpecies() sage: c = P.generating_series()[0:3] sage: P._check() True sage: P == loads(dumps(P)) True - >>> from sage.all import * >>> P = species.PermutationSpecies() >>> P.generating_series()[Integer(0):Integer(5)] [1, 1, 1, 1, 1] >>> P.isotype_generating_series()[Integer(0):Integer(5)] [1, 1, 2, 3, 5] >>> P = species.PermutationSpecies() >>> c = P.generating_series()[Integer(0):Integer(3)] >>> P._check() True >>> P == loads(dumps(P)) True 
- class sage.combinat.species.permutation_species.PermutationSpeciesStructure(parent, labels, list)[source]¶
- Bases: - GenericSpeciesStructure- automorphism_group()[source]¶
- Return the group of permutations whose action on this structure leave it fixed. - EXAMPLES: - sage: set_random_seed(0) sage: p = PermutationGroupElement((2,3,4)) sage: P = species.PermutationSpecies() sage: a = P.structures(["a", "b", "c", "d"])[2]; a ['a', 'c', 'b', 'd'] sage: a.automorphism_group() Permutation Group with generators [(2,3), (1,4)] - >>> from sage.all import * >>> set_random_seed(Integer(0)) >>> p = PermutationGroupElement((Integer(2),Integer(3),Integer(4))) >>> P = species.PermutationSpecies() >>> a = P.structures(["a", "b", "c", "d"])[Integer(2)]; a ['a', 'c', 'b', 'd'] >>> a.automorphism_group() Permutation Group with generators [(2,3), (1,4)] - sage: [a.transport(perm) for perm in a.automorphism_group()] [['a', 'c', 'b', 'd'], ['a', 'c', 'b', 'd'], ['a', 'c', 'b', 'd'], ['a', 'c', 'b', 'd']] - >>> from sage.all import * >>> [a.transport(perm) for perm in a.automorphism_group()] [['a', 'c', 'b', 'd'], ['a', 'c', 'b', 'd'], ['a', 'c', 'b', 'd'], ['a', 'c', 'b', 'd']] 
 - canonical_label()[source]¶
- EXAMPLES: - sage: P = species.PermutationSpecies() sage: S = P.structures(["a", "b", "c"]) sage: [s.canonical_label() for s in S] [['a', 'b', 'c'], ['b', 'a', 'c'], ['b', 'a', 'c'], ['b', 'c', 'a'], ['b', 'c', 'a'], ['b', 'a', 'c']] - >>> from sage.all import * >>> P = species.PermutationSpecies() >>> S = P.structures(["a", "b", "c"]) >>> [s.canonical_label() for s in S] [['a', 'b', 'c'], ['b', 'a', 'c'], ['b', 'a', 'c'], ['b', 'c', 'a'], ['b', 'c', 'a'], ['b', 'a', 'c']] 
 - permutation_group_element()[source]¶
- Return - selfas a permutation group element.- EXAMPLES: - sage: p = PermutationGroupElement((2,3,4)) sage: P = species.PermutationSpecies() sage: a = P.structures(["a", "b", "c", "d"])[2]; a ['a', 'c', 'b', 'd'] sage: a.permutation_group_element() (2,3) - >>> from sage.all import * >>> p = PermutationGroupElement((Integer(2),Integer(3),Integer(4))) >>> P = species.PermutationSpecies() >>> a = P.structures(["a", "b", "c", "d"])[Integer(2)]; a ['a', 'c', 'b', 'd'] >>> a.permutation_group_element() (2,3) 
 - transport(perm)[source]¶
- Return the transport of this structure along the permutation perm. - EXAMPLES: - sage: p = PermutationGroupElement((2,3,4)) sage: P = species.PermutationSpecies() sage: a = P.structures(["a", "b", "c", "d"])[2]; a ['a', 'c', 'b', 'd'] sage: a.transport(p) ['a', 'd', 'c', 'b'] - >>> from sage.all import * >>> p = PermutationGroupElement((Integer(2),Integer(3),Integer(4))) >>> P = species.PermutationSpecies() >>> a = P.structures(["a", "b", "c", "d"])[Integer(2)]; a ['a', 'c', 'b', 'd'] >>> a.transport(p) ['a', 'd', 'c', 'b'] 
 
- sage.combinat.species.permutation_species.PermutationSpecies_class[source]¶
- alias of - PermutationSpecies