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EXAMPLE
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The a(3) = 16 antichains of multisets:
(111),
(122), (12)(22), (1)(22),
(112), (11)(12), (2)(11),
(123), (13)(23), (12)(23), (12)(13), (12)(13)(23), (3)(12), (2)(13), (1)(23), (1)(2)(3).
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MATHEMATICA
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stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==w||Q[r, w]||Q[w, r]], Q]]]];
multijoin[mss__]:=Join@@Table[Table[x, {Max[Count[#, x]&/@{mss}]}], {x, Union[mss]}]
submultisetQ[M_, N_]:=Or[Length[M]==0, MatchQ[{Sort[List@@M], Sort[List@@N]}, {{x_, Z___}, {___, x_, W___}}/; submultisetQ[{Z}, {W}]]];
allnorm[n_]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
auu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]], submultisetQ], multijoin@@#==m&];
Table[Length[Join@@Table[auu[m], {m, allnorm[n]}]], {n, 5}]
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