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 A320330 Number of T_0 multiset partitions of integer partitions of n. 8
 1, 1, 3, 5, 13, 25, 50, 100, 195, 366, 707, 1333, 2440 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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}}. The T_0 condition means the dual is strict. LINKS EXAMPLE The a(1) = 1 through a(5) = 25 multiset partitions:   {{1}}  {{2}}      {{3}}          {{4}}              {{5}}          {{1,1}}    {{1,1,1}}      {{2,2}}            {{1,1,3}}          {{1},{1}}  {{1},{2}}      {{1,1,2}}          {{1,2,2}}                     {{1},{1,1}}    {{1},{3}}          {{1},{4}}                     {{1},{1},{1}}  {{2},{2}}          {{2},{3}}                                    {{1,1,1,1}}        {{1,1,1,2}}                                    {{1},{1,2}}        {{1},{1,3}}                                    {{2},{1,1}}        {{1},{2,2}}                                    {{1},{1,1,1}}      {{2},{1,2}}                                    {{1,1},{1,1}}      {{3},{1,1}}                                    {{1},{1},{2}}      {{1,1,1,1,1}}                                    {{1},{1},{1,1}}    {{1},{1,1,2}}                                    {{1},{1},{1},{1}}  {{1,1},{1,2}}                                                       {{1},{1},{3}}                                                       {{1},{2},{2}}                                                       {{2},{1,1,1}}                                                       {{1},{1,1,1,1}}                                                       {{1,1},{1,1,1}}                                                       {{1},{1},{1,2}}                                                       {{1},{2},{1,1}}                                                       {{1},{1},{1,1,1}}                                                       {{1},{1,1},{1,1}}                                                       {{1},{1},{1},{2}}                                                       {{1},{1},{1},{1,1}}                                                       {{1},{1},{1},{1},{1}} 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]]]]; dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}]; Table[Length[Select[Join@@mps/@IntegerPartitions[n], UnsameQ@@dual[#]&]], {n, 8}] CROSSREFS Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A316983, A319066, A319312, A320328, A320331. Sequence in context: A026766 A026709 A219699 * A159290 A110494 A098615 Adjacent sequences:  A320327 A320328 A320329 * A320331 A320332 A320333 KEYWORD nonn,more AUTHOR Gus Wiseman, Oct 11 2018 STATUS approved

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Last modified November 27 07:28 EST 2021. Contains 349365 sequences. (Running on oeis4.)