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%I #6 Aug 18 2019 11:27:48
%S 1,1,3,21,913,1183295
%N Number of pairwise intersecting set-systems covering n vertices where every two vertices appear together in some edge (cointersecting).
%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}}. This sequence counts pairwise intersecting, covering set-systems that are cointersecting, meaning their dual is pairwise intersecting.
%F Inverse binomial transform of A327038.
%e The a(0) = 1 through a(3) = 21 set-systems:
%e {} {{1}} {{1,2}} {{1,2,3}}
%e {{1},{1,2}} {{1},{1,2,3}}
%e {{2},{1,2}} {{2},{1,2,3}}
%e {{3},{1,2,3}}
%e {{1,2},{1,2,3}}
%e {{1,3},{1,2,3}}
%e {{2,3},{1,2,3}}
%e {{1},{1,2},{1,2,3}}
%e {{1},{1,3},{1,2,3}}
%e {{1,2},{1,3},{2,3}}
%e {{2},{1,2},{1,2,3}}
%e {{2},{2,3},{1,2,3}}
%e {{3},{1,3},{1,2,3}}
%e {{3},{2,3},{1,2,3}}
%e {{1,2},{1,3},{1,2,3}}
%e {{1,2},{2,3},{1,2,3}}
%e {{1,3},{2,3},{1,2,3}}
%e {{1},{1,2},{1,3},{1,2,3}}
%e {{2},{1,2},{2,3},{1,2,3}}
%e {{3},{1,3},{2,3},{1,2,3}}
%e {{1,2},{1,3},{2,3},{1,2,3}}
%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[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]
%Y Intersecting covering set-systems are A305843.
%Y The unlabeled multiset partition version is A319765.
%Y The case where the dual is strict is A319774.
%Y The BII-numbers of these set-systems are A326912.
%Y The non-covering version is A327038.
%Y Cointersectng covering set-systems are A327040.
%Y Cf. A003465, A319767, A326853, A326854, A326910.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Aug 17 2019
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