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

0,5

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.

Links

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

Example

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}}

Prog

(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

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.

Sequence in context: A079904 A175630 A021420 * A198543 A367468 A294846

Adjacent sequences: A330470 A330471 A330472 * A330474 A330475 A330476

Keyword

nonn,tabl

Author

Gus Wiseman, Dec 20 2019

Extensions

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

Status

approved