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A320353
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Number of antichains of multisets whose multiset union is an integer partition of n.
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7
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1, 1, 3, 5, 11, 17, 36, 56, 107, 175, 311, 505, 887
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OFFSET
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0,3
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LINKS
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EXAMPLE
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The a(1) = 1 through a(5) = 17 antichains:
{{1}} {{2}} {{3}} {{4}} {{5}}
{{1,1}} {{1,2}} {{1,3}} {{1,4}}
{{1},{1}} {{1,1,1}} {{2,2}} {{2,3}}
{{1},{2}} {{1,1,2}} {{1,1,3}}
{{1},{1},{1}} {{1},{3}} {{1,2,2}}
{{2},{2}} {{1},{4}}
{{1,1,1,1}} {{2},{3}}
{{2},{1,1}} {{1,1,1,2}}
{{1,1},{1,1}} {{1},{2,2}}
{{1},{1},{2}} {{3},{1,1}}
{{1},{1},{1},{1}} {{1,1,1,1,1}}
{{1,1},{1,2}}
{{1},{1},{3}}
{{1},{2},{2}}
{{2},{1,1,1}}
{{1},{1},{1},{2}}
{{1},{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]]]];
submultisetQ[M_, N_]:=Or[Length[M]==0, MatchQ[{Sort[List@@M], Sort[List@@N]}, {{x_, Z___}, {___, x_, W___}}/; submultisetQ[{Z}, {W}]]];
antiQ[s_]:=Select[Tuples[s, 2], And[UnsameQ@@#, submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n], antiQ]], {n, 8}]
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CROSSREFS
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Cf. A001970, A006126, A050342, A089259, A258466, A261005, A261049, A293994, A319719, A319721, A320328, A320355, A320356.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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