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# 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)

    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)

  Bases: `sage.combinat.species.structure.SpeciesStructureWrapper`

  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

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