Examples of Combinatorial Species#
- sage.combinat.species.library.BinaryForestSpecies()#
Return 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()[: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]]
- sage.combinat.species.library.BinaryTreeSpecies()#
Return the species of binary trees on \(n\) leaves.
The species of binary trees \(B\) is defined by \(B = X + B \cdot B\), 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])[187]; a 2*((5*3)*(4*1)) sage: a.automorphism_group() Permutation Group with generators [()]
- sage.combinat.species.library.SimpleGraphSpecies()#
Return 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()[: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()[:6] [1, 1, 2, 4, 11, 34]