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EXAMPLE
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The A255906(5) - a(5) = 22 crossing multiset partitions:
{{13}{124}} {{1}{13}{24}}
{{13}{224}} {{1}{24}{35}}
{{13}{234}} {{2}{13}{24}}
{{13}{244}} {{2}{14}{35}}
{{13}{245}} {{3}{13}{24}}
{{14}{235}} {{3}{14}{25}}
{{24}{113}} {{4}{13}{24}}
{{24}{123}} {{4}{13}{25}}
{{24}{133}} {{5}{13}{24}}
{{24}{134}}
{{24}{135}}
{{25}{134}}
{{35}{124}}
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MATHEMATICA
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nonXQ[stn_]:=!MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];
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[Sum[Length[Select[mps[m], nonXQ]], {m, allnorm[n]}], {n, 0, 8}]
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