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%I #6 Oct 12 2018 22:43:07
%S 1,1,3,5,11,17,36,56,107,175,311,505,887
%N Number of antichains of multisets whose multiset union is an integer partition of n.
%H Goran Kilibarda and Vladeta Jovovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Kilibarda/kili2.pdf">Antichains of Multisets</a>, Journal of Integer Sequences, Vol. 7 (2004).
%e The a(1) = 1 through a(5) = 17 antichains:
%e {{1}} {{2}} {{3}} {{4}} {{5}}
%e {{1,1}} {{1,2}} {{1,3}} {{1,4}}
%e {{1},{1}} {{1,1,1}} {{2,2}} {{2,3}}
%e {{1},{2}} {{1,1,2}} {{1,1,3}}
%e {{1},{1},{1}} {{1},{3}} {{1,2,2}}
%e {{2},{2}} {{1},{4}}
%e {{1,1,1,1}} {{2},{3}}
%e {{2},{1,1}} {{1,1,1,2}}
%e {{1,1},{1,1}} {{1},{2,2}}
%e {{1},{1},{2}} {{3},{1,1}}
%e {{1},{1},{1},{1}} {{1,1,1,1,1}}
%e {{1,1},{1,2}}
%e {{1},{1},{3}}
%e {{1},{2},{2}}
%e {{2},{1,1,1}}
%e {{1},{1},{1},{2}}
%e {{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],antiQ]],{n,8}]
%Y Cf. A001970, A006126, A050342, A089259, A258466, A261005, A261049, A293994, A319719, A319721, A320328, A320355, A320356.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Oct 11 2018
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