# Recursive Species¶

class sage.combinat.species.recursive_species.CombinatorialSpecies

EXAMPLES:

sage: F = CombinatorialSpecies()
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.generating_series().coefficients(4)
[1, 1, 1, 1]
define(x)

Defines 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]]

TESTS:

sage: A = CombinatorialSpecies()
sage: A.define(E+X*A*A)
sage: A.generating_series().coefficients(6)
[1, 1, 2, 5, 14, 42]
sage: A.generating_series().counts(6)
[1, 1, 4, 30, 336, 5040]
sage: len(A.structures([1,2,3,4]).list())
336
sage: A.isotype_generating_series().coefficients(6)
[1, 1, 2, 5, 14, 42]
sage: len(A.isotypes([1,2,3,4]).list())
14
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)

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)
True

Empty Species

#### Next topic

Characteristic Species