%I #7 Jul 19 2019 07:51:55

%S 1,0,1,5,61,2721,788221

%N Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having different sums.

%C An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

%e The a(3) = 5 antichains:

%e {{1,2,3}}

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

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

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

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

%e The a(4) = 61 antichains:

%e {1234} {12}{34} {12}{13}{14} {12}{13}{14}{24} {12}{13}{14}{24}{34}

%e {13}{24} {12}{13}{24} {12}{13}{14}{34} {12}{13}{23}{24}{34}

%e {12}{134} {12}{13}{34} {12}{13}{23}{24}

%e {12}{234} {12}{14}{34} {12}{13}{23}{34}

%e {13}{124} {12}{23}{24} {12}{13}{24}{34}

%e {13}{234} {12}{23}{34} {12}{14}{24}{34}

%e {14}{123} {12}{24}{34} {12}{23}{24}{34}

%e {14}{234} {13}{14}{24} {13}{14}{24}{34}

%e {23}{124} {13}{23}{24} {13}{23}{24}{34}

%e {23}{134} {13}{23}{34} {12}{13}{14}{234}

%e {24}{134} {13}{24}{34} {12}{23}{24}{134}

%e {34}{123} {14}{24}{34} {123}{124}{134}{234}

%e {123}{124} {12}{13}{234}

%e {123}{134} {12}{14}{234}

%e {123}{234} {12}{23}{134}

%e {124}{134} {12}{24}{134}

%e {124}{234} {13}{14}{234}

%e {134}{234} {13}{23}{124}

%e {14}{34}{123}

%e {23}{24}{134}

%e {12}{134}{234}

%e {13}{124}{234}

%e {14}{123}{234}

%e {23}{124}{134}

%e {123}{124}{134}

%e {123}{124}{234}

%e {123}{134}{234}

%e {124}{134}{234}

%t stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];

%t cleq[n_]:=Select[stableSets[Subsets[Range[n],{2,n}],SubsetQ[#1,#2]||Total[#1]==Total[#2]&],Union@@#==Range[n]&];

%t Table[Length[cleq[n]],{n,0,5}]

%Y Antichain covers are A006126.

%Y Set partitions with different block-sums are A275780.

%Y MM-numbers of multiset partitions with different part-sums are A326535.

%Y Antichain covers with equal edge-sums and no singletons are A326565.

%Y Antichain covers with different edge-sizes and no singletons are A326569.

%Y The case with singletons allowed is A326572.

%Y Antichains with equal edge-sums are A326574.

%Y Cf. A000372, A003182, A035470, A307249, A321469, A326519, A326566, A326570, A326573.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Jul 18 2019