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%I #11 Aug 14 2019 17:46:48
%S 1,0,1,1,4,13,91,1318,73581,51913025
%N Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having the same sum.
%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(2) = 1 through a(5) = 13 antichains:
%e {{1,2}} {{1,2,3}} {{1,2,3,4}} {{1,2,3,4,5}}
%e {{1,4},{2,3}} {{1,2,5},{1,3,4}}
%e {{2,4},{1,2,3}} {{1,3,5},{2,3,4}}
%e {{3,4},{1,2,4}} {{1,4,5},{2,3,5}}
%e {{1,4,5},{1,2,3,4}}
%e {{2,3,5},{1,2,3,4}}
%e {{2,4,5},{1,2,3,5}}
%e {{3,4,5},{1,2,4,5}}
%e {{1,5},{2,4},{1,2,3}}
%e {{2,5},{3,4},{1,2,4}}
%e {{3,5},{1,2,5},{1,3,4}}
%e {{4,5},{1,3,5},{2,3,4}}
%e {{1,4,5},{2,3,5},{1,2,3,4}}
%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 Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326360, A326518, A326534, A326566.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Jul 13 2019
%E a(9) from _Andrew Howroyd_, Aug 14 2019
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