A318227 Number of inequivalent leaf-colorings of rooted identity trees with n nodes.

1, 1, 1, 3, 5, 14, 38, 114, 330, 1054, 3483, 11841, 41543, 149520, 552356, 2084896, 8046146, 31649992, 127031001, 518434863, 2153133594, 9081863859, 38909868272, 169096646271, 745348155211, 3329032020048, 15063018195100, 68998386313333, 319872246921326, 1500013368166112

Offset

1,4

Comments

In a rooted identity tree, all branches directly under any given branch are different.

The leaves are colored after selection of the tree. Since all trees are asymmetric, the symmetry group of the leaves will be the identity group and a tree with k leaves will have Bell(k) inequivalent leaf-colorings. - Andrew Howroyd, Dec 10 2020

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Formula

a(n) = Sum_{k=1..n} A055327(n,k) * A000110(k). - Andrew Howroyd, Dec 10 2020

Example

Inequivalent representatives of the a(6) = 14 leaf-colorings:
  (1(1(1)))  ((1)((1)))  (1(((1))))  ((1((1))))  (((1(1))))  (((((1)))))
  (1(1(2)))  ((1)((2)))  (1(((2))))  ((1((2))))  (((1(2))))
  (1(2(1)))
  (1(2(2)))
  (1(2(3)))

Mathematica

idt[n_]:=If[n==1, {{}}, Join@@Table[Select[Union[Sort/@Tuples[idt/@c]], UnsameQ@@#&], {c, IntegerPartitions[n-1]}]];
Table[Sum[BellB[Count[tree, {}, {0, Infinity}]], {tree, idt[n]}], {n, 16}]

Prog

(PARI) \\ bell(n) is A000110(n).
WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i ))-1)}
bell(n)={sum(k=1, n, stirling(n, k, 2))}
seq(n)={my(v=[y], b=vector(n, k, bell(k))); for(n=2, n, v=concat(v[1], WeighMT(v))); vector(n, k, sum(i=1, k, polcoef(v[k], i)*b[i]))} \\ Andrew Howroyd, Dec 10 2020

Crossrefs

Cf. A000081, A001190, A001678, A003238, A004111, A290689, A318185, A304486.

Cf. A318226, A318228, A318229, A318230, A318231, A318234.

Cf. A000110 (Bell numbers), A055327, A301342.

Sequence in context: A295064 A052974 A284415 * A363542 A230585 A006395

Adjacent sequences: A318224 A318225 A318226 * A318228 A318229 A318230

Keyword

nonn

Author

Gus Wiseman, Aug 21 2018

Extensions

Terms a(17) and beyond from Andrew Howroyd, Dec 10 2020

Status

approved