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EXAMPLE
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The a(0) = 1 through a(4) = 15 normal multiset partitions:
{} {{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
{{1,2}} {{1,1,2}} {{1,1,1,2}}
{{1},{1}} {{1,2,2}} {{1,1,2,2}}
{{1,2,3}} {{1,1,2,3}}
{{2},{1,1}} {{1,2,2,2}}
{{3},{1,2}} {{1,2,2,3}}
{{1},{1},{1}} {{1,2,3,3}}
{{1,2,3,4}}
{{1,1},{1,1}}
{{1,2},{1,2}}
{{1,3},{2,2}}
{{1,4},{2,3}}
{{2},{2},{1,1}}
{{3},{3},{1,2}}
{{1},{1},{1},{1}}
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MATHEMATICA
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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]]]];
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n], SameQ@@Total/@#&]], {n, 0, 5}]
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