%I #6 Oct 12 2018 22:43:20

%S 1,1,2,3,5,8,13,22,35,62,98,171,277

%N Number of strict connected 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(6) = 13 clutters:

%e {{1}} {{2}} {{3}} {{4}} {{5}} {{6}}

%e {{1,1}} {{1,2}} {{1,3}} {{1,4}} {{1,5}}

%e {{1,1,1}} {{2,2}} {{2,3}} {{2,4}}

%e {{1,1,2}} {{1,1,3}} {{3,3}}

%e {{1,1,1,1}} {{1,2,2}} {{1,1,4}}

%e {{1,1,1,2}} {{1,2,3}}

%e {{1,1,1,1,1}} {{2,2,2}}

%e {{1,1},{1,2}} {{1,1,1,3}}

%e {{1,1,2,2}}

%e {{1,1,1,1,2}}

%e {{1,1},{1,3}}

%e {{1,1,1,1,1,1}}

%e {{1,2},{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[UnsameQ@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,8}]

%Y Cf. A001970, A007718, A048143, A056156, A258466, A261006, A293994, A318403, A319079, A319719, A319721, A320351, A320353, A320355.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Oct 11 2018