Permutation species

sage.combinat.species.permutation_species.PermutationSpecies(*args, **kwds)

Returns the species of permutations.

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: P.generating_series().coefficients(5)
[1, 1, 1, 1, 1]
sage: P.isotype_generating_series().coefficients(5)
[1, 1, 2, 3, 5]
class sage.combinat.species.permutation_species.PermutationSpeciesStructure(parent, labels, list)
automorphism_group()

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

EXAMPLES:

sage: p = PermutationGroupElement((2,3,4))
sage: P = species.PermutationSpecies()
sage: a = P.structures(["a", "b", "c", "d"]).random_element(); a
['a', 'c', 'b', 'd']
sage: 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']]
canonical_label()

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']]
permutation_group_element()

Returns self as a permutation group element.

EXAMPLES:

sage: p = PermutationGroupElement((2,3,4))
sage: P = species.PermutationSpecies()
sage: a = P.structures(["a", "b", "c", "d"]).random_element(); a
['a', 'c', 'b', 'd']
sage: a.permutation_group_element()
(2,3)
transport(perm)

Returns 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"]).random_element(); a
['a', 'c', 'b', 'd']
sage: a.transport(p)
['a', 'd', 'c', 'b']
class sage.combinat.species.permutation_species.PermutationSpecies_class(min=None, max=None, weight=None)
__init__(min=None, max=None, weight=None)

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: c = P.generating_series().coefficients(3)
sage: P._check()
True
sage: P == loads(dumps(P))
True
static _cached_constructor(*args, **kwds)

Returns the species of permutations.

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: P.generating_series().coefficients(5)
[1, 1, 1, 1, 1]
sage: P.isotype_generating_series().coefficients(5)
[1, 1, 2, 3, 5]
_canonical_rep_from_partition(structure_class, labels, p)

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: P._canonical_rep_from_partition(P._default_structure_class, ["a","b","c"], [2,1])
['b', 'a', 'c']
_cis(series_ring, base_ring)

The cycle index series for the species of permutations is given by

\prod{n=1}^\infty \frac{1}{1-x_n}.

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: g = P.cycle_index_series()
sage: g.coefficients(5)
[p[],
 p[1],
 p[1, 1] + p[2],
 p[1, 1, 1] + p[2, 1] + p[3],
 p[1, 1, 1, 1] + p[2, 1, 1] + p[2, 2] + p[3, 1] + p[4]]
_cis_gen(base_ring, n)

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: g = P._cis_gen(QQ, 2)
sage: [g.next() for i in range(10)]
[p[], 0, p[2], 0, p[2, 2], 0, p[2, 2, 2], 0, p[2, 2, 2, 2], 0]
_default_structure_class
alias of PermutationSpeciesStructure
_gs_list(base_ring)

The generating series for the species of linear orders is \frac{1}{1-x}.

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: g = P.generating_series()
sage: g.coefficients(10)
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
_isotypes(structure_class, labels)

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: P.isotypes([1,2,3]).list()
[[2, 3, 1], [2, 1, 3], [1, 2, 3]]
_itgs_iterator(base_ring)

The isomorphism type generating series is given by \frac{1}{1-x}.

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: g = P.isotype_generating_series()
sage: g.coefficients(10)
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]
_structures(structure_class, labels)

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: P.structures([1,2,3]).list()
[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]

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