A318566 Number of non-isomorphic multiset partitions of multiset partitions of multisets of size n.

1, 6, 21, 104, 452, 2335, 11992, 66810, 385101, 2336352, 14738380, 96831730, 659809115, 4657075074, 33974259046, 255781455848, 1984239830571, 15839628564349, 129951186405574, 1094486382191624, 9453318070371926, 83654146992936350, 757769011659766015, 7020652591448497490

Offset: 1

Author

Gus Wiseman, Aug 29 2018

Keywords

• nonn

Examples

			Non-isomorphic representatives of the a(3) = 21 multiset partitions of multiset partitions:
  {{{1,1,1}}}
  {{{1,1,2}}}
  {{{1,2,3}}}
  {{{1},{1,1}}}
  {{{1},{1,2}}}
  {{{1},{2,3}}}
  {{{2},{1,1}}}
  {{{1},{1},{1}}}
  {{{1},{1},{2}}}
  {{{1},{2},{3}}}
  {{{1}},{{1,1}}}
  {{{1}},{{1,2}}}
  {{{1}},{{2,3}}}
  {{{2}},{{1,1}}}
  {{{1}},{{1},{1}}}
  {{{1}},{{1},{2}}}
  {{{1}},{{2},{3}}}
  {{{2}},{{1},{1}}}
  {{{1}},{{1}},{{1}}}
  {{{1}},{{1}},{{2}}}
  {{{1}},{{2}},{{3}}}
		

Crossrefs

Cf. A001970, A007716, A050336, A050338, A255906, A269134, A317533, A317791, A318564, A318565.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
dubnorm[m_]:=First[Union[Table[Map[Sort,m/.Rule@@@Table[{Union[Flatten[m]][[i]],Union[Flatten[m]][[perm[[i]]]]},{i,Length[perm]}],{0,2}],{perm,Permutations[Union[Flatten[m]]]}]]];
Table[Length[Union[dubnorm/@Join@@mps/@Join@@mps/@strnorm[n]]],{n,5}]

PARI

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=sExp(symGroupSeries(n))); NumUnlabeledObjsSeq(sCartProd(A, sExp(A)-1))} \\ Andrew Howroyd, Dec 30 2020

Extensions

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020