A320665 Number of non-isomorphic multiset partitions of weight n with no singletons or vertices that appear only once.

1, 0, 1, 1, 5, 6, 27, 47, 169, 406, 1327, 3790, 12560, 39919, 136821, 470589, 1687981, 6162696, 23173374, 88981796, 349969596, 1405386733, 5764142220, 24111709328, 102825231702, 446665313598, 1975339030948, 8888051121242, 40667889052853, 189126710033882, 893526261542899

Offset: 0

Author

Gus Wiseman, Oct 18 2018

Keywords

• nonn

Comments

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. This sequence counts non-isomorphic multiset partitions with no singletons whose dual also has no singletons.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 27 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}    {{1,1,1,1,1,1}}
                      {{1,1,2,2}}    {{1,1,2,2,2}}    {{1,1,1,2,2,2}}
                      {{1,1},{1,1}}  {{1,1},{1,1,1}}  {{1,1,2,2,2,2}}
                      {{1,1},{2,2}}  {{1,1},{1,2,2}}  {{1,1,2,2,3,3}}
                      {{1,2},{1,2}}  {{1,1},{2,2,2}}  {{1,1},{1,1,1,1}}
                                     {{1,2},{1,2,2}}  {{1,1,1},{1,1,1}}
                                                      {{1,1},{1,2,2,2}}
                                                      {{1,1,1},{2,2,2}}
                                                      {{1,1,2},{1,2,2}}
                                                      {{1,1},{2,2,2,2}}
                                                      {{1,1,2},{2,2,2}}
                                                      {{1,1},{2,2,3,3}}
                                                      {{1,1,2},{2,3,3}}
                                                      {{1,2},{1,1,2,2}}
                                                      {{1,2},{1,2,2,2}}
                                                      {{1,2},{1,2,3,3}}
                                                      {{1,2,2},{1,2,2}}
                                                      {{1,2,3},{1,2,3}}
                                                      {{2,2},{1,1,2,2}}
                                                      {{1,1},{1,1},{1,1}}
                                                      {{1,1},{1,2},{2,2}}
                                                      {{1,1},{2,2},{2,2}}
                                                      {{1,1},{2,2},{3,3}}
                                                      {{1,1},{2,3},{2,3}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,2},{2,2}}
                                                      {{1,2},{1,3},{2,3}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

Row sums of A369287.

Cf. A001055, A007716, A306005, A317794, A318871, A320663, A320664, A321760.

Programs

PARI

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

PARI

Vec(G(20,1)) \\ G defined in A369287. - Andrew Howroyd, Jan 28 2024

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023