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

0,3

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.

Links

Table of n, a(n) for n=0..25.

Example

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}}

Prog

(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

Crossrefs

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

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

Sequence in context: A135958 A265787 A163665 * A215381 A215377 A051079

Adjacent sequences: A323783 A323784 A323785 * A323787 A323788 A323789

Keyword

nonn

Author

Gus Wiseman, Jan 28 2019

Extensions

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

Status

approved