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)
__repr__()

EXAMPLES:

sage: S = species.CycleSpecies()
sage: a = S.structures(["a","b","c"]).random_element(); a
('a', 'b', 'c')
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)
__init__(min=None, max=None, weight=None)

EXAMPLES:

sage: P = species.CycleSpecies()
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 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
_cis_iterator(base_ring)

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

-\sum_{k=1}^\infty \phi(k)/k * log(1 - x_k)

which is equal to

\sum_{n=1}^\infty \frac{1}{n} * \sum_{k|n} \phi(k) * x_k^{n/k}

.

EXAMPLES:

sage: P = species.CycleSpecies()
sage: cis = P.cycle_index_series()
sage: cis.coefficients(7)
[0,
 p[1],
 1/2*p[1, 1] + 1/2*p[2],
 1/3*p[1, 1, 1] + 2/3*p[3],
 1/4*p[1, 1, 1, 1] + 1/4*p[2, 2] + 1/2*p[4],
 1/5*p[1, 1, 1, 1, 1] + 4/5*p[5],
 1/6*p[1, 1, 1, 1, 1, 1] + 1/6*p[2, 2, 2] + 1/3*p[3, 3] + 1/3*p[6]]
_default_structure_class
alias of CycleSpeciesStructure
_gs_iterator(base_ring)

The generating series for cyclic permutations is -\log(1-x) = \sum_{n=1}^\infty x^n/n.

EXAMPLES:

sage: P = species.CycleSpecies()
sage: g = P.generating_series()
sage: g.coefficients(10)
[0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9]

TESTS:

sage: P = species.CycleSpecies()
sage: g = P.generating_series(RR)
sage: g.coefficients(3)
[0.000000000000000, 1.00000000000000, 0.500000000000000]
_isotypes(structure_class, labels)

EXAMPLES:

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

The isomorphism type generating series for cyclic permutations is given by x/(1-x).

EXAMPLES:

sage: P = species.CycleSpecies()
sage: g = P.isotype_generating_series()
sage: g.coefficients(5)
[0, 1, 1, 1, 1]

TESTS:

sage: P = species.CycleSpecies()
sage: g = P.isotype_generating_series(RR)
sage: g.coefficients(3)
[0.000000000000000, 1.00000000000000, 1.00000000000000]
_order()

Returns the order of the generating series.

EXAMPLES:

sage: P = species.CycleSpecies()
sage: P._order()
1
_structures(structure_class, labels)

EXAMPLES:

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

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