Cycle Species#

class sage.combinat.species.cycle_species.CycleSpecies(min=None, max=None, weight=None)[source]#

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)]

>>> 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]#
automorphism_group()[source]#

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])[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]#

Returns 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]#

Returns 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