Linear-order species - Combinatorics

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# Linear-order species

class sage.combinat.species.linear_order_species.LinearOrderSpecies(min=None, max=None, weight=None)

  Bases: `GenericCombinatorialSpecies`, `UniqueRepresentation`

  Return 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)

  Bases: `GenericSpeciesStructure`

  automorphism_group()

    Return 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()

    EXAMPLES:

    Sage

        sage: P = species.LinearOrderSpecies()
        doctest:warning...
        DeprecationWarning: combinat.species is superseded by LazyCombinatorialSpecies
        See https://github.com/sagemath/sage/issues/38544 for details.
        sage: s = P.structures(["a", "b", "c"]).random_element()
        sage: s.canonical_label()
        ['a', 'b', 'c']

    Python

        >>> from sage.all import *
        >>> P = species.LinearOrderSpecies()
        doctest:warning...
        DeprecationWarning: combinat.species is superseded by LazyCombinatorialSpecies
        See https://github.com/sagemath/sage/issues/38544 for details.
        >>> s = P.structures(["a", "b", "c"]).random_element()
        >>> s.canonical_label()
        ['a', 'b', 'c']

  transport(perm)

    Return 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

  alias of `LinearOrderSpecies`