Subset Species

class sage.combinat.species.subset_species.SubsetSpecies(min=None, max=None, weight=None)

Bases: GenericCombinatorialSpecies, UniqueRepresentation

Return the species of subsets.

EXAMPLES:

Sage

sage: S = species.SubsetSpecies()
sage: S.generating_series()[0:5]
[1, 2, 2, 4/3, 2/3]
sage: S.isotype_generating_series()[0:5]
[1, 2, 3, 4, 5]

sage: S = species.SubsetSpecies()
sage: c = S.generating_series()[0:3]
sage: S._check()
True
sage: S == loads(dumps(S))
True

Python

>>> from sage.all import *
>>> S = species.SubsetSpecies()
>>> S.generating_series()[Integer(0):Integer(5)]
[1, 2, 2, 4/3, 2/3]
>>> S.isotype_generating_series()[Integer(0):Integer(5)]
[1, 2, 3, 4, 5]

>>> S = species.SubsetSpecies()
>>> c = S.generating_series()[Integer(0):Integer(3)]
>>> S._check()
True
>>> S == loads(dumps(S))
True

class sage.combinat.species.subset_species.SubsetSpeciesStructure(parent, labels, list)

Bases: GenericSpeciesStructure

automorphism_group()

Return the group of permutations whose action on this subset leave it fixed.

EXAMPLES:

Sage

sage: F = species.SubsetSpecies()
sage: a = F.structures([1,2,3,4])[6]; a
{1, 3}
sage: a.automorphism_group()                                                # needs sage.groups
Permutation Group with generators [(2,4), (1,3)]

Python

>>> from sage.all import *
>>> F = species.SubsetSpecies()
>>> a = F.structures([Integer(1),Integer(2),Integer(3),Integer(4)])[Integer(6)]; a
{1, 3}
>>> a.automorphism_group()                                                # needs sage.groups
Permutation Group with generators [(2,4), (1,3)]

Sage

sage: [a.transport(g) for g in a.automorphism_group()]                      # needs sage.groups
[{1, 3}, {1, 3}, {1, 3}, {1, 3}]

Python

>>> from sage.all import *
>>> [a.transport(g) for g in a.automorphism_group()]                      # needs sage.groups
[{1, 3}, {1, 3}, {1, 3}, {1, 3}]

canonical_label()

Return the canonical label of self.

EXAMPLES:

Sage

sage: P = species.SubsetSpecies()
sage: S = P.structures(["a", "b", "c"])
sage: [s.canonical_label() for s in S]
[{}, {'a'}, {'a'}, {'a'}, {'a', 'b'}, {'a', 'b'}, {'a', 'b'}, {'a', 'b', 'c'}]

Python

>>> from sage.all import *
>>> P = species.SubsetSpecies()
>>> S = P.structures(["a", "b", "c"])
>>> [s.canonical_label() for s in S]
[{}, {'a'}, {'a'}, {'a'}, {'a', 'b'}, {'a', 'b'}, {'a', 'b'}, {'a', 'b', 'c'}]

complement()

Return the complement of self.

EXAMPLES:

Sage

sage: F = species.SubsetSpecies()
sage: a = F.structures(["a", "b", "c"])[5]; a
{'a', 'c'}
sage: a.complement()
{'b'}

Python

>>> from sage.all import *
>>> F = species.SubsetSpecies()
>>> a = F.structures(["a", "b", "c"])[Integer(5)]; a
{'a', 'c'}
>>> a.complement()
{'b'}

label_subset()

Return a subset of the labels that “appear” in this structure.

EXAMPLES:

Sage

sage: P = species.SubsetSpecies()
sage: S = P.structures(["a", "b", "c"])
sage: [s.label_subset() for s in S]
[[], ['a'], ['b'], ['c'], ['a', 'b'], ['a', 'c'], ['b', 'c'], ['a', 'b', 'c']]

Python

>>> from sage.all import *
>>> P = species.SubsetSpecies()
>>> S = P.structures(["a", "b", "c"])
>>> [s.label_subset() for s in S]
[[], ['a'], ['b'], ['c'], ['a', 'b'], ['a', 'c'], ['b', 'c'], ['a', 'b', 'c']]

transport(perm)

Return the transport of this subset along the permutation perm.

EXAMPLES:

Sage

sage: F = species.SubsetSpecies()
sage: a = F.structures(["a", "b", "c"])[5]; a
{'a', 'c'}
sage: p = PermutationGroupElement((1,2))                                    # needs sage.groups
sage: a.transport(p)                                                        # needs sage.groups
{'b', 'c'}
sage: p = PermutationGroupElement((1,3))                                    # needs sage.groups
sage: a.transport(p)                                                        # needs sage.groups
{'a', 'c'}

Python

>>> from sage.all import *
>>> F = species.SubsetSpecies()
>>> a = F.structures(["a", "b", "c"])[Integer(5)]; a
{'a', 'c'}
>>> p = PermutationGroupElement((Integer(1),Integer(2)))                                    # needs sage.groups
>>> a.transport(p)                                                        # needs sage.groups
{'b', 'c'}
>>> p = PermutationGroupElement((Integer(1),Integer(3)))                                    # needs sage.groups
>>> a.transport(p)                                                        # needs sage.groups
{'a', 'c'}

sage.combinat.species.subset_species.SubsetSpecies_class

alias of SubsetSpecies