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%I #18 Oct 14 2018 09:16:40
%S 1,1,2,3,4,4,8,7,12,15,19,26,43
%N Number of connected antichains of sets whose multiset union is an integer partition of n.
%e The a(10) = 19 clutters:
%e {{10}}
%e {{1,9}}
%e {{2,8}}
%e {{3,7}}
%e {{4,6}}
%e {{1,2,7}}
%e {{1,3,6}}
%e {{1,4,5}}
%e {{2,3,5}}
%e {{1,2,3,4}}
%e {{5},{5}}
%e {{1,2},{1,6}}
%e {{1,2},{2,5}}
%e {{1,3},{1,5}}
%e {{1,4},{1,4}}
%e {{2,3},{2,3}}
%e {{1,2},{1,2},{1,3}}
%e {{2},{2},{2},{2},{2}}
%e {{1},{1},{1},{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 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%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@@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,10}]
%Y Cf. A001970, A007718, A048143, A056156, A089259, A319719, A320351, A320353, A320355, A320356.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Oct 12 2018
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