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%I #5 Oct 14 2018 09:16:30
%S 1,1,2,4,6,9,18,24,39,58,92,131,206
%N Number of antichains of sets whose multiset union is an integer partition of n.
%e The a(1) = 1 through a(7) = 24 antichains:
%e {{1}} {{2}} {{3}} {{4}} {{5}}
%e {{1},{1}} {{1,2}} {{1,3}} {{1,4}}
%e {{1},{2}} {{1},{3}} {{2,3}}
%e {{1},{1},{1}} {{2},{2}} {{1},{4}}
%e {{1},{1},{2}} {{2},{3}}
%e {{1},{1},{1},{1}} {{1},{1},{3}}
%e {{1},{2},{2}}
%e {{1},{1},{1},{2}}
%e {{1},{1},{1},{1},{1}}
%e .
%e {{6}} {{7}}
%e {{1,5}} {{1,6}}
%e {{2,4}} {{2,5}}
%e {{1,2,3}} {{3,4}}
%e {{1},{5}} {{1,2,4}}
%e {{2},{4}} {{1},{6}}
%e {{3},{3}} {{2},{5}}
%e {{1},{2,3}} {{3},{4}}
%e {{2},{1,3}} {{1},{2,4}}
%e {{3},{1,2}} {{2},{1,4}}
%e {{1},{1},{4}} {{4},{1,2}}
%e {{1,2},{1,2}} {{1},{1},{5}}
%e {{1},{2},{3}} {{1,2},{1,3}}
%e {{2},{2},{2}} {{1},{2},{4}}
%e {{1},{1},{1},{3}} {{1},{3},{3}}
%e {{1},{1},{2},{2}} {{2},{2},{3}}
%e {{1},{1},{1},{1},{2}} {{1},{1},{2,3}}
%e {{1},{1},{1},{1},{1},{1}} {{1},{1},{1},{4}}
%e {{1},{1},{2},{3}}
%e {{1},{2},{2},{2}}
%e {{1},{1},{1},{1},{3}}
%e {{1},{1},{1},{2},{2}}
%e {{1},{1},{1},{1},{1},{2}}
%e {{1},{1},{1},{1},{1},{1},{1}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
%t Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[And@@UnsameQ@@@#,antiQ[#]]&]],{n,10}]
%Y Cf. A001970, A089259, A258466, A319719, A319721, A320328, A320353, A320355, A320356.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Oct 12 2018
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