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Number of strict antichains of sets whose multiset union is an integer partition of n.
0

%I #5 Oct 14 2018 09:16:50

%S 1,1,1,3,3,5,10,13,19,28,47,64,98

%N Number of strict antichains of sets whose multiset union is an integer partition of n.

%e The a(1) = 1 through a(8) = 19 antichains:

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

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

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

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

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

%e {{2},{4}}

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

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

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

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

%e .

%e {{7}} {{8}}

%e {{1,6}} {{1,7}}

%e {{2,5}} {{2,6}}

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

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

%e {{1},{6}} {{1,3,4}}

%e {{2},{5}} {{1},{7}}

%e {{3},{4}} {{2},{6}}

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

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

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

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

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

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

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

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

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

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

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

%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[UnsameQ@@#,And@@UnsameQ@@@#,antiQ[#]]&]],{n,10}]

%Y Cf. A001970, A050342, A089259, A258466, A261049, A319719, A319721, A320328, A320331, A320353.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Oct 12 2018