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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

Table of n, a(n) for n=0..12.

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.)