Linear-order Species#

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

Bases: GenericCombinatorialSpecies, UniqueRepresentation

Returns the species of linear orders.

EXAMPLES:

Sage

sage: L = species.LinearOrderSpecies()
sage: L.generating_series()[0:5]
[1, 1, 1, 1, 1]

sage: L = species.LinearOrderSpecies()
sage: L._check()
True
sage: L == loads(dumps(L))
True

Python

>>> from sage.all import *
>>> L = species.LinearOrderSpecies()
>>> L.generating_series()[Integer(0):Integer(5)]
[1, 1, 1, 1, 1]

>>> L = species.LinearOrderSpecies()
>>> L._check()
True
>>> L == loads(dumps(L))
True

class sage.combinat.species.linear_order_species.LinearOrderSpeciesStructure(parent, labels, list)[source]#

Bases: GenericSpeciesStructure

automorphism_group()[source]#

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

sage: F = species.LinearOrderSpecies()
sage: a = F.structures(["a", "b", "c"])[0]; a
['a', 'b', 'c']
sage: a.automorphism_group()                                                # needs sage.groups
Symmetric group of order 1! as a permutation group

Python

>>> from sage.all import *
>>> F = species.LinearOrderSpecies()
>>> a = F.structures(["a", "b", "c"])[Integer(0)]; a
['a', 'b', 'c']
>>> a.automorphism_group()                                                # needs sage.groups
Symmetric group of order 1! as a permutation group

canonical_label()[source]#

EXAMPLES:

Sage

sage: P = species.LinearOrderSpecies()
sage: s = P.structures(["a", "b", "c"]).random_element()
sage: s.canonical_label()
['a', 'b', 'c']

Python

>>> from sage.all import *
>>> P = species.LinearOrderSpecies()
>>> s = P.structures(["a", "b", "c"]).random_element()
>>> s.canonical_label()
['a', 'b', 'c']

transport(perm)[source]#

Returns the transport of this structure along the permutation perm.

EXAMPLES:

Sage

sage: F = species.LinearOrderSpecies()
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
['b', 'a', 'c']

Python

>>> from sage.all import *
>>> F = species.LinearOrderSpecies()
>>> 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
['b', 'a', 'c']

sage.combinat.species.linear_order_species.LinearOrderSpecies_class[source]#: alias of LinearOrderSpecies