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%I #4 Aug 18 2019 11:27:01
%S 1,1,1,3,155
%N Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain.
%C A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.
%F Inverse binomial transform of A327059.
%e The a(0) = 1 through a(3) = 3 set-systems:
%e {} {{1}} {{12}} {{123}}
%e {{12}{13}{23}}
%e {{12}{13}{23}{123}}
%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%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 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],Union@@#==Range[n]&&stableQ[dual[#],SubsetQ]&]],{n,0,3}]
%Y Covering intersecting set-systems are A305843.
%Y The BII-numbers of these set-systems are the intersection of A326910 and A326966.
%Y Covering coantichains are A326970.
%Y The non-covering version is A327059.
%Y The unlabeled multiset partition version is A327060.
%Y Cf. A006126, A051185, A059523, A305844, A319639, A326961, A326965, A326968, A327020, A327057.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Aug 18 2019
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