Recursive Species

class sage.combinat.species.recursive_species.CombinatorialSpecies

Bases: sage.combinat.species.species.GenericCombinatorialSpecies

EXAMPLES:

sage: F = CombinatorialSpecies()
sage: loads(dumps(F))
Combinatorial species
sage: X = species.SingletonSpecies()
sage: E = species.EmptySetSpecies()
sage: L = CombinatorialSpecies()
sage: L.define(E+X*L)
sage: L.generating_series().coefficients(4)
[1, 1, 1, 1]
sage: LL = loads(dumps(L))
sage: LL.generating_series().coefficients(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: X = species.SingletonSpecies()
sage: E = species.EmptySetSpecies()
sage: L = CombinatorialSpecies()
sage: L.define(E+X*L)
sage: L.generating_series().coefficients(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*{}))]
sage: L = species.LinearOrderSpecies()
sage: L.generating_series().coefficients(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]]
sage: A = CombinatorialSpecies()
sage: A.define(X+A*A)
sage: A.generating_series().coefficients(6)
[0, 1, 1, 2, 5, 14]
sage: A.generating_series().counts(6)
[0, 1, 2, 12, 120, 1680]
sage: len(A.structures([1,2,3]).list())
12
sage: A.isotype_generating_series().coefficients(6)
[0, 1, 1, 2, 5, 14]
sage: len(A.isotypes([1,2,3,4]).list())
5
sage: X2 = X*X
sage: X5 = X2*X2*X
sage: A = CombinatorialSpecies()
sage: B = CombinatorialSpecies()
sage: C = CombinatorialSpecies()
sage: A.define(X5+B*B)
sage: B.define(X5+C*C)
sage: C.define(X2+C*C+A*A)
sage: A.generating_series().coefficients(Integer(10))
[0, 0, 0, 0, 0, 1, 0, 0, 1, 2]
sage: A.generating_series().coefficients(15)
[0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 5, 4, 14, 10, 48]
sage: B.generating_series().coefficients(15)
[0, 0, 0, 0, 1, 1, 2, 0, 5, 0, 14, 0, 44, 0, 138]
sage: C.generating_series().coefficients(15)
[0, 0, 1, 0, 1, 0, 2, 0, 5, 0, 15, 0, 44, 2, 142]
sage: F = CombinatorialSpecies()
sage: F.define(E+X+(X*F+X*X*F))
sage: F.generating_series().counts(10)
[1, 2, 6, 30, 192, 1560, 15120, 171360, 2217600, 32296320]
sage: F.generating_series().coefficients(10)
[1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
sage: F.isotype_generating_series().coefficients(10)
[1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
weight_ring()

EXAMPLES:

sage: F = species.CombinatorialSpecies()
sage: F.weight_ring()
Rational Field
sage: X = species.SingletonSpecies()
sage: E = species.EmptySetSpecies()
sage: L = CombinatorialSpecies()
sage: L.define(E+X*L)
sage: L.weight_ring()
Rational Field
class sage.combinat.species.recursive_species.CombinatorialSpeciesStructure(parent, s, **options)

Bases: sage.combinat.species.structure.SpeciesStructureWrapper