A330473 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.

1, 0, 1, 0, 2, 4, 0, 3, 8, 10, 0, 5, 28, 38, 33, 0, 7, 56, 146, 152, 91, 0, 11, 138, 474, 786, 628, 298, 0, 15, 268, 1388, 3117, 3808, 2486, 910, 0, 22, 570, 3843, 11830, 19147, 18395, 9986, 3017, 0, 30, 1072, 10094, 40438, 87081, 110164, 86388, 39889, 9945

Offset: 0

Author

Gus Wiseman, Dec 20 2019

Keywords

• nonn
• tabl

Comments

As an alternative description, T(n,k) is the number of non-isomorphic multisets of nonempty multisets of nonempty multisets with n leaves whose multiset union consists of k multisets.

Examples

			Triangle begins:
   1
   0   1
   0   2   4
   0   3   8  10
   0   5  28  38  33
   0   7  56 146 152  91
   0  11 138 474 786 628 298
For example, row n = 3 counts the following multiset partitions:
  {{111}}  {{1}{11}}    {{1}{1}{1}}
  {{112}}  {{1}{12}}    {{1}{1}{2}}
  {{123}}  {{1}{23}}    {{1}{2}{3}}
           {{2}{11}}    {{1}}{{1}{1}}
           {{1}}{{11}}  {{1}}{{1}{2}}
           {{1}}{{12}}  {{1}}{{2}{3}}
           {{1}}{{23}}  {{2}}{{1}{1}}
           {{2}}{{11}}  {{1}}{{1}}{{1}}
                        {{1}}{{1}}{{2}}
                        {{1}}{{2}}{{3}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..350

Crossrefs

Row sums are A318566.

Column k = 1 is A000041 (for n > 0).

Column k = n is A007716.

Partitions of partitions of partitions are A007713.

Twice-factorizations are A050336.

The 2-dimensional version is A317533.

See A330472 for a variation.

Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.

Programs

PARI

\\ See links in A339645 for combinatorial species functions.
ColGf(k, n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(sExp(A), k, x)*x^k + O(x*x^n), A) ))}
M(n, m=n)={Mat(vector(m+1, k, Col(ColGf(k-1, n), -(n+1))))}
{ my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 18 2023

Extensions

Terms a(36) and beyond from Andrew Howroyd, Jan 18 2023