A320328 - OEIS

A320328

Number of square multiset partitions of integer partitions of n.

1, 1, 2, 3, 6, 11, 20, 36, 65, 117, 214, 382, 679

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OFFSET

0,3

COMMENTS

A multiset partition is square if its length is equal to its number of distinct atoms.

LINKS

EXAMPLE

The a(1) = 1 through a(6) = 20 square partitions:

{{1,1}} {{1,1,1}} {{2,2}} {{1},{4}} {{3,3}}

{{1},{2}} {{1},{3}} {{2},{3}} {{1},{5}}

{{1,1,1,1}} {{1},{1,3}} {{2,2,2}}

{{1},{1,2}} {{1},{2,2}} {{2},{4}}

{{2},{1,1}} {{2},{1,2}} {{1},{1,4}}

{{3},{1,1}} {{4},{1,1}}

{{1,1,1,1,1}} {{1},{1,1,3}}

{{1},{1,1,2}} {{1,1},{1,3}}

{{1,1},{1,2}} {{1},{1,2,2}}

{{2},{1,1,1}} {{1,1},{2,2}}

{{1,2},{1,2}}

{{1},{2},{3}}

{{2},{1,1,2}}

{{3},{1,1,1}}

{{1},{1,1,1,2}}

{{1,1},{1,1,2}}

{{1,2},{1,1,1}}

{{2},{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]]]];

Table[Length[Select[Join@@mps/@IntegerPartitions[n], Length[#]==Length[Union@@#]&]], {n, 8}]

CROSSREFS

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Oct 11 2018

STATUS

approved