Returns the species of linear orders.
EXAMPLES:
sage: L = species.LinearOrderSpecies()
sage: L.generating_series().coefficients(5)
[1, 1, 1, 1, 1]
Returns the group of permutations whose action on this structure leave it fixed. For the species of linear orders, there is no non-trivial automorphism.
EXAMPLES:
sage: F = species.LinearOrderSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
['a', 'b', 'c']
sage: a.automorphism_group()
Symmetric group of order 1! as a permutation group
EXAMPLES:
sage: P = species.LinearOrderSpecies()
sage: s = P.structures(["a", "b", "c"]).random_element()
sage: s.canonical_label()
['a', 'b', 'c']
Returns the transport of this structure along the permutation perm.
EXAMPLES:
sage: F = species.LinearOrderSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
['a', 'b', 'c']
sage: p = PermutationGroupElement((1,2))
sage: a.transport(p)
['b', 'a', 'c']
EXAMPLES:
sage: L = species.LinearOrderSpecies()
sage: L._check()
True
sage: L == loads(dumps(L))
True
Returns the species of linear orders.
EXAMPLES:
sage: L = species.LinearOrderSpecies()
sage: L.generating_series().coefficients(5)
[1, 1, 1, 1, 1]
EXAMPLES:
sage: L = species.LinearOrderSpecies()
sage: g = L.cycle_index_series()
sage: g.coefficients(5)
[p[], p[1], p[1, 1], p[1, 1, 1], p[1, 1, 1, 1]]
The generating series for the species of linear orders is
.
EXAMPLES:
sage: L = species.LinearOrderSpecies()
sage: g = L.generating_series()
sage: g.coefficients(10)
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
EXAMPLES:
sage: L = species.LinearOrderSpecies()
sage: L.isotypes([1,2,3]).list()
[[1, 2, 3]]
The isomorphism type generating series is given by
.
EXAMPLES:
sage: L = species.LinearOrderSpecies()
sage: g = L.isotype_generating_series()
sage: g.coefficients(10)
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
EXAMPLES:
sage: L = species.LinearOrderSpecies()
sage: L.structures([1,2,3]).list()
[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]