A323786 Number of non-isomorphic weight-n multisets of multisets of non-singleton multisets.

1, 0, 2, 3, 19, 39, 200, 615, 2849, 11174, 52377, 239269, 1191090, 6041975, 32275288, 177797719, 1017833092, 6014562272, 36717301665, 230947360981, 1495562098099, 9956230757240, 68070158777759, 477439197541792, 3432259679880648, 25267209686664449

Offset: 0

Author

Gus Wiseman, Jan 28 2019

Keywords

• nonn

Comments

All sets and multisets must be finite, and only the outermost may be empty.

The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(4) = 19 multiset partitions:
  {{1111}}      {{1112}}      {{1123}}      {{1234}}
  {{11}{11}}    {{1122}}      {{11}{23}}    {{12}{34}}
  {{11}}{{11}}  {{11}{12}}    {{12}{13}}    {{12}}{{34}}
                {{11}{22}}    {{11}}{{23}}
                {{12}{12}}    {{12}}{{13}}
                {{11}}{{12}}
                {{11}}{{22}}
                {{12}}{{12}}
Non-isomorphic representatives of the a(5) = 39 multiset partitions:
  {{11111}}      {{11112}}      {{11123}}      {{11234}}      {{12345}}
  {{11}{111}}    {{11122}}      {{11223}}      {{11}{234}}    {{12}{345}}
  {{11}}{{111}}  {{11}{112}}    {{11}{123}}    {{12}{134}}    {{12}}{{345}}
                 {{11}{122}}    {{11}{223}}    {{23}{114}}
                 {{12}{111}}    {{12}{113}}    {{11}}{{234}}
                 {{12}{112}}    {{12}{123}}    {{12}}{{134}}
                 {{22}{111}}    {{13}{122}}    {{23}}{{114}}
                 {{11}}{{112}}  {{23}{111}}
                 {{11}}{{122}}  {{11}}{{123}}
                 {{12}}{{111}}  {{11}}{{223}}
                 {{12}}{{112}}  {{12}}{{113}}
                 {{22}}{{111}}  {{12}}{{123}}
                                {{13}}{{122}}
                                {{23}}{{111}}
		

Crossrefs

Cf. A007716, A302545, A306186, A317791, A318564, A318566.

Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323794.

Programs

PARI

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp(sExp(A-x*sv(1)))))} \\ Andrew Howroyd, Jan 17 2023

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jan 17 2023