Recursive Species

class sage.combinat.species.recursive_species.CombinatorialSpecies(min=None)

Bases: GenericCombinatorialSpecies

EXAMPLES:

Sage

sage: F = CombinatorialSpecies()
sage: loads(dumps(F))
Combinatorial species

Python

>>> from sage.all import *
>>> F = CombinatorialSpecies()
>>> loads(dumps(F))
Combinatorial species

Sage

sage: X = species.SingletonSpecies()
sage: E = species.EmptySetSpecies()
sage: L = CombinatorialSpecies()
sage: L.define(E+X*L)
sage: L.generating_series()[0:4]
[1, 1, 1, 1]
sage: LL = loads(dumps(L))
sage: LL.generating_series()[0:4]
[1, 1, 1, 1]

Python

>>> from sage.all import *
>>> X = species.SingletonSpecies()
>>> E = species.EmptySetSpecies()
>>> L = CombinatorialSpecies()
>>> L.define(E+X*L)
>>> L.generating_series()[Integer(0):Integer(4)]
[1, 1, 1, 1]
>>> LL = loads(dumps(L))
>>> LL.generating_series()[Integer(0):Integer(4)]
[1, 1, 1, 1]

define(x)

Define self to be equal to the combinatorial species x.

This is used to define combinatorial species recursively. All of the real work is done by calling the .set() method for each of the series associated to self.

EXAMPLES: The species of linear orders L can be recursively defined by \(L = 1 + X*L\) where 1 represents the empty set species and X represents the singleton species.

Sage

sage: X = species.SingletonSpecies()
sage: E = species.EmptySetSpecies()
sage: L = CombinatorialSpecies()
sage: L.define(E+X*L)
sage: L.generating_series()[0:4]
[1, 1, 1, 1]
sage: L.structures([1,2,3]).cardinality()
6
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*{}))]

Python

>>> from sage.all import *
>>> X = species.SingletonSpecies()
>>> E = species.EmptySetSpecies()
>>> L = CombinatorialSpecies()
>>> L.define(E+X*L)
>>> L.generating_series()[Integer(0):Integer(4)]
[1, 1, 1, 1]
>>> L.structures([Integer(1),Integer(2),Integer(3)]).cardinality()
6
>>> L.structures([Integer(1),Integer(2),Integer(3)]).list()
[1*(2*(3*{})),
 1*(3*(2*{})),
 2*(1*(3*{})),
 2*(3*(1*{})),
 3*(1*(2*{})),
 3*(2*(1*{}))]

Sage

sage: L = species.LinearOrderSpecies()
sage: L.generating_series()[0:4]
[1, 1, 1, 1]
sage: L.structures([1,2,3]).cardinality()
6
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]]

Python

>>> from sage.all import *
>>> L = species.LinearOrderSpecies()
>>> L.generating_series()[Integer(0):Integer(4)]
[1, 1, 1, 1]
>>> L.structures([Integer(1),Integer(2),Integer(3)]).cardinality()
6
>>> L.structures([Integer(1),Integer(2),Integer(3)]).list()
[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]

weight_ring()

EXAMPLES:

Sage

sage: F = species.CombinatorialSpecies()
sage: F.weight_ring()
Rational Field

Python

>>> from sage.all import *
>>> F = species.CombinatorialSpecies()
>>> F.weight_ring()
Rational Field

Sage

sage: X = species.SingletonSpecies()
sage: E = species.EmptySetSpecies()
sage: L = CombinatorialSpecies()
sage: L.define(E+X*L)
sage: L.weight_ring()
Rational Field

Python

>>> from sage.all import *
>>> X = species.SingletonSpecies()
>>> E = species.EmptySetSpecies()
>>> L = CombinatorialSpecies()
>>> L.define(E+X*L)
>>> L.weight_ring()
Rational Field

class sage.combinat.species.recursive_species.CombinatorialSpeciesStructure(parent, s, **options)

Bases: SpeciesStructureWrapper