### Navigation

- index
- modules |
- next |
- previous |
- Combinatorics »
- Combinatorial Species »

# Examples of Combinatorial Species

`sage.combinat.species.library.``BinaryForestSpecies`()

  Returns the species of binary forests. Binary forests are defined as sets of binary trees.

  EXAMPLES:

      sage: F = species.BinaryForestSpecies()
      sage: F.generating_series().counts(10)
      [1, 1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353]
      sage: F.isotype_generating_series().counts(10)
      [1, 1, 2, 4, 10, 26, 77, 235, 758, 2504]
      sage: F.cycle_index_series().coefficients(7)
      [p[],
       p[1],
       3/2*p[1, 1] + 1/2*p[2],
       19/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3],
       193/24*p[1, 1, 1, 1] + 3/4*p[2, 1, 1] + 5/8*p[2, 2] + 1/3*p[3, 1] + 1/4*p[4],
       907/40*p[1, 1, 1, 1, 1] + 19/12*p[2, 1, 1, 1] + 5/8*p[2, 2, 1] + 1/2*p[3, 1, 1] + 1/6*p[3, 2] + 1/4*p[4, 1] + 1/5*p[5],
       49171/720*p[1, 1, 1, 1, 1, 1] + 193/48*p[2, 1, 1, 1, 1] + 15/16*p[2, 2, 1, 1] + 61/48*p[2, 2, 2] + 19/18*p[3, 1, 1, 1] + 1/6*p[3, 2, 1] + 7/18*p[3, 3] + 3/8*p[4, 1, 1] + 1/8*p[4, 2] + 1/5*p[5, 1] + 1/6*p[6]]

  TESTS:

      sage: seq = F.isotype_generating_series().counts(10)[1:] #optional
      sage: oeis(seq)[0]                              # optional -- internet
      A052854: Number of forests of ordered trees on n total nodes.

`sage.combinat.species.library.``BinaryTreeSpecies`()

  Returns the species of binary trees on n leaves. The species of binary trees B is defined by B = X + B*B where X is the singleton species.

  EXAMPLES:

      sage: B = species.BinaryTreeSpecies()
      sage: B.generating_series().counts(10)
      [0, 1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400]
      sage: B.isotype_generating_series().counts(10)
      [0, 1, 1, 2, 5, 14, 42, 132, 429, 1430]
      sage: B._check()
      True

      sage: B = species.BinaryTreeSpecies()
      sage: a = B.structures([1,2,3,4,5]).random_element(); a
      2*((5*3)*(4*1))
      sage: a.automorphism_group()
      Permutation Group with generators [()]

  TESTS:

      sage: seq = B.isotype_generating_series().counts(10)[1:] #optional
      sage: oeis(seq)[0]                              # optional -- internet
      A000108: Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.

`sage.combinat.species.library.``SimpleGraphSpecies`()

  Returns the species of simple graphs.

  EXAMPLES:

      sage: S = species.SimpleGraphSpecies()
      sage: S.generating_series().counts(10)
      [1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456, 68719476736]
      sage: S.cycle_index_series().coefficients(5)
      [p[],
       p[1],
       p[1, 1] + p[2],
       4/3*p[1, 1, 1] + 2*p[2, 1] + 2/3*p[3],
       8/3*p[1, 1, 1, 1] + 4*p[2, 1, 1] + 2*p[2, 2] + 4/3*p[3, 1] + p[4]]
      sage: S.isotype_generating_series().coefficients(6)
      [1, 1, 2, 4, 11, 34]

  TESTS:

      sage: seq = S.isotype_generating_series().counts(6)[1:]  #optional
      sage: oeis(seq)[0]                              # optional -- internet
      A000088: Number of graphs on n unlabeled nodes.

      sage: seq = S.generating_series().counts(10)[1:]  #optional
      sage: oeis(seq)[0]                              # optional -- internet
      A006125: 2^(n(n-1)/2).

### Navigation

- index
- modules |
- next |
- previous |
- Combinatorics »
- Combinatorial Species »