Species structures#
We will illustrate the use of the structure classes using the “balls and bars” model for integer compositions. An integer composition of 6 such as [2, 1, 3] can be represented in this model as ‘oooooo’ where the 6 o’s correspond to the balls and the 2 ‘s correspond to the bars. If BB is our species for this model, the it satisfies the following recursive definition:
BB = o + o*BB + o*|*BB
Here we define this species using the default structures:
sage: ball = species.SingletonSpecies()
sage: bar = species.EmptySetSpecies()
sage: BB = CombinatorialSpecies()
sage: BB.define(ball + ball*BB + ball*bar*BB)
sage: o = var('o') # needs sage.symbolic
sage: BB.isotypes([o]*3).list() # needs sage.symbolic
[o*(o*o), o*((o*{})*o), (o*{})*(o*o), (o*{})*((o*{})*o)]
>>> from sage.all import *
>>> ball = species.SingletonSpecies()
>>> bar = species.EmptySetSpecies()
>>> BB = CombinatorialSpecies()
>>> BB.define(ball + ball*BB + ball*bar*BB)
>>> o = var('o') # needs sage.symbolic
>>> BB.isotypes([o]*Integer(3)).list() # needs sage.symbolic
[o*(o*o), o*((o*{})*o), (o*{})*(o*o), (o*{})*((o*{})*o)]
If we ignore the parentheses, we can read off that the integer compositions are [3], [2, 1], [1, 2], and [1, 1, 1].
- class sage.combinat.species.structure.GenericSpeciesStructure(parent, labels, list)[source]#
Bases:
CombinatorialObject
This is a base class from which the classes for the structures inherit.
EXAMPLES:
sage: from sage.combinat.species.structure import GenericSpeciesStructure sage: a = GenericSpeciesStructure(None, [2,3,4], [1,2,3]) sage: a [2, 3, 4] sage: a.parent() is None True sage: a == loads(dumps(a)) True
>>> from sage.all import * >>> from sage.combinat.species.structure import GenericSpeciesStructure >>> a = GenericSpeciesStructure(None, [Integer(2),Integer(3),Integer(4)], [Integer(1),Integer(2),Integer(3)]) >>> a [2, 3, 4] >>> a.parent() is None True >>> a == loads(dumps(a)) True
- change_labels(labels)[source]#
Return a relabelled structure.
INPUT:
labels
, a list of labels.
OUTPUT:
A structure with the i-th label of self replaced with the i-th label of the list.
EXAMPLES:
sage: P = species.SubsetSpecies() sage: S = P.structures(["a", "b", "c"]) sage: [s.change_labels([1,2,3]) for s in S] [{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}]
>>> from sage.all import * >>> P = species.SubsetSpecies() >>> S = P.structures(["a", "b", "c"]) >>> [s.change_labels([Integer(1),Integer(2),Integer(3)]) for s in S] [{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}]
- is_isomorphic(x)[source]#
EXAMPLES:
sage: S = species.SetSpecies() sage: a = S.structures([1,2,3]).random_element(); a {1, 2, 3} sage: b = S.structures(['a','b','c']).random_element(); b {'a', 'b', 'c'} sage: a.is_isomorphic(b) True
>>> from sage.all import * >>> S = species.SetSpecies() >>> a = S.structures([Integer(1),Integer(2),Integer(3)]).random_element(); a {1, 2, 3} >>> b = S.structures(['a','b','c']).random_element(); b {'a', 'b', 'c'} >>> a.is_isomorphic(b) True
- labels()[source]#
Returns the labels used for this structure.
Note
This includes labels which may not “appear” in this particular structure.
EXAMPLES:
sage: P = species.SubsetSpecies() sage: s = P.structures(["a", "b", "c"]).random_element() sage: s.labels() ['a', 'b', 'c']
>>> from sage.all import * >>> P = species.SubsetSpecies() >>> s = P.structures(["a", "b", "c"]).random_element() >>> s.labels() ['a', 'b', 'c']
- parent()[source]#
Returns the species that this structure is associated with.
EXAMPLES:
sage: L = species.LinearOrderSpecies() sage: a,b = L.structures([1,2]) sage: a.parent() Linear order species
>>> from sage.all import * >>> L = species.LinearOrderSpecies() >>> a,b = L.structures([Integer(1),Integer(2)]) >>> a.parent() Linear order species
- class sage.combinat.species.structure.IsotypesWrapper(species, labels, structure_class)[source]#
Bases:
SpeciesWrapper
A base class for the set of isotypes of a species with given set of labels. An object of this type is returned when you call the
isotypes()
method of a species.EXAMPLES:
sage: F = species.SetSpecies() sage: S = F.isotypes([1,2,3]) sage: S == loads(dumps(S)) True
>>> from sage.all import * >>> F = species.SetSpecies() >>> S = F.isotypes([Integer(1),Integer(2),Integer(3)]) >>> S == loads(dumps(S)) True
- class sage.combinat.species.structure.SimpleIsotypesWrapper(species, labels, structure_class)[source]#
Bases:
SpeciesWrapper
Warning
This is deprecated and currently not used for anything.
EXAMPLES:
sage: F = species.SetSpecies() sage: S = F.structures([1,2,3]) sage: S == loads(dumps(S)) True
>>> from sage.all import * >>> F = species.SetSpecies() >>> S = F.structures([Integer(1),Integer(2),Integer(3)]) >>> S == loads(dumps(S)) True
- class sage.combinat.species.structure.SimpleStructuresWrapper(species, labels, structure_class)[source]#
Bases:
SpeciesWrapper
Warning
This is deprecated and currently not used for anything.
EXAMPLES:
sage: F = species.SetSpecies() sage: S = F.structures([1,2,3]) sage: S == loads(dumps(S)) True
>>> from sage.all import * >>> F = species.SetSpecies() >>> S = F.structures([Integer(1),Integer(2),Integer(3)]) >>> S == loads(dumps(S)) True
- sage.combinat.species.structure.SpeciesStructure[source]#
alias of
GenericSpeciesStructure
- class sage.combinat.species.structure.SpeciesStructureWrapper(parent, s, **options)[source]#
Bases:
GenericSpeciesStructure
This is a class for the structures of species such as the sum species that do not provide “additional” structure. For example, if you have the sum \(C\) of species \(A\) and \(B\), then a structure of \(C\) will either be either something from \(A\) or \(B\). Instead of just returning one of these directly, a “wrapper” is put around them so that they have their parent is \(C\) rather than \(A\) or \(B\):
sage: X = species.SingletonSpecies() sage: X2 = X+X sage: s = X2.structures([1]).random_element(); s 1 sage: s.parent() Sum of (Singleton species) and (Singleton species) sage: from sage.combinat.species.structure import SpeciesStructureWrapper sage: issubclass(type(s), SpeciesStructureWrapper) True
>>> from sage.all import * >>> X = species.SingletonSpecies() >>> X2 = X+X >>> s = X2.structures([Integer(1)]).random_element(); s 1 >>> s.parent() Sum of (Singleton species) and (Singleton species) >>> from sage.combinat.species.structure import SpeciesStructureWrapper >>> issubclass(type(s), SpeciesStructureWrapper) True
EXAMPLES:
sage: E = species.SetSpecies(); B = E+E sage: s = B.structures([1,2,3]).random_element() sage: s.parent() Sum of (Set species) and (Set species) sage: s == loads(dumps(s)) True
>>> from sage.all import * >>> E = species.SetSpecies(); B = E+E >>> s = B.structures([Integer(1),Integer(2),Integer(3)]).random_element() >>> s.parent() Sum of (Set species) and (Set species) >>> s == loads(dumps(s)) True
- canonical_label()[source]#
EXAMPLES:
sage: P = species.PartitionSpecies() sage: s = (P+P).structures([1,2,3])[1]; s # needs sage.libs.flint {{1, 3}, {2}} sage: s.canonical_label() # needs sage.libs.flint {{1, 2}, {3}}
>>> from sage.all import * >>> P = species.PartitionSpecies() >>> s = (P+P).structures([Integer(1),Integer(2),Integer(3)])[Integer(1)]; s # needs sage.libs.flint {{1, 3}, {2}} >>> s.canonical_label() # needs sage.libs.flint {{1, 2}, {3}}
- change_labels(labels)[source]#
Return a relabelled structure.
INPUT:
labels
, a list of labels.
OUTPUT:
A structure with the i-th label of self replaced with the i-th label of the list.
EXAMPLES:
sage: X = species.SingletonSpecies() sage: X2 = X+X sage: s = X2.structures([1]).random_element(); s 1 sage: s.change_labels(['a']) 'a'
>>> from sage.all import * >>> X = species.SingletonSpecies() >>> X2 = X+X >>> s = X2.structures([Integer(1)]).random_element(); s 1 >>> s.change_labels(['a']) 'a'
- transport(perm)[source]#
EXAMPLES:
sage: P = species.PartitionSpecies() sage: s = (P+P).structures([1,2,3])[1]; s # needs sage.libs.flint {{1, 3}, {2}} sage: s.transport(PermutationGroupElement((2,3))) # needs sage.groups sage.libs.flint {{1, 2}, {3}}
>>> from sage.all import * >>> P = species.PartitionSpecies() >>> s = (P+P).structures([Integer(1),Integer(2),Integer(3)])[Integer(1)]; s # needs sage.libs.flint {{1, 3}, {2}} >>> s.transport(PermutationGroupElement((Integer(2),Integer(3)))) # needs sage.groups sage.libs.flint {{1, 2}, {3}}
- class sage.combinat.species.structure.SpeciesWrapper(species, labels, iterator, generating_series, name, structure_class)[source]#
Bases:
Parent
This is a abstract base class for the set of structures of a species as well as the set of isotypes of the species.
Note
One typically does not use
SpeciesWrapper
directly, but instead instantiates one of its subclasses:StructuresWrapper
orIsotypesWrapper
.EXAMPLES:
sage: from sage.combinat.species.structure import SpeciesWrapper sage: F = species.SetSpecies() sage: S = SpeciesWrapper(F, [1,2,3], "_structures", "generating_series", 'Structures', None) sage: S Structures for Set species with labels [1, 2, 3] sage: S.list() [{1, 2, 3}] sage: S.cardinality() 1
>>> from sage.all import * >>> from sage.combinat.species.structure import SpeciesWrapper >>> F = species.SetSpecies() >>> S = SpeciesWrapper(F, [Integer(1),Integer(2),Integer(3)], "_structures", "generating_series", 'Structures', None) >>> S Structures for Set species with labels [1, 2, 3] >>> S.list() [{1, 2, 3}] >>> S.cardinality() 1
- cardinality()[source]#
Returns the number of structures in this set.
EXAMPLES:
sage: F = species.SetSpecies() sage: F.structures([1,2,3]).cardinality() 1
>>> from sage.all import * >>> F = species.SetSpecies() >>> F.structures([Integer(1),Integer(2),Integer(3)]).cardinality() 1
- labels()[source]#
Returns the labels used on these structures. If \(X\) is the species, then
labels()
returns the preimage of these structures under the functor \(X\).EXAMPLES:
sage: F = species.SetSpecies() sage: F.structures([1,2,3]).labels() [1, 2, 3]
>>> from sage.all import * >>> F = species.SetSpecies() >>> F.structures([Integer(1),Integer(2),Integer(3)]).labels() [1, 2, 3]
- class sage.combinat.species.structure.StructuresWrapper(species, labels, structure_class)[source]#
Bases:
SpeciesWrapper
A base class for the set of structures of a species with given set of labels. An object of this type is returned when you call the
structures()
method of a species.EXAMPLES:
sage: F = species.SetSpecies() sage: S = F.structures([1,2,3]) sage: S == loads(dumps(S)) True
>>> from sage.all import * >>> F = species.SetSpecies() >>> S = F.structures([Integer(1),Integer(2),Integer(3)]) >>> S == loads(dumps(S)) True