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%I #9 Oct 22 2023 16:49:34
%S 1,1,4,72,25104,2077196832,9221293229809363008,
%T 170141182628636920877978969957369949312
%N Number of set-systems covering n vertices, every two of which 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 covering set-systems that are cointersecting, meaning their dual is pairwise intersecting.
%F Inverse binomial transform of A327039.
%e The a(0) = 1 through a(2) = 4 set-systems:
%e {} {{1}} {{1,2}}
%e {{1},{1,2}}
%e {{2},{1,2}}
%e {{1},{2},{1,2}}
%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]
%Y The unlabeled multiset partition version is A319752.
%Y The BII-numbers of these set-systems are A326853.
%Y The antichain case is A327020.
%Y The pairwise intersecting case is A327037.
%Y The non-covering version is A327039.
%Y The case where the dual is strict is A327053.
%Y Cf. A003465, A305843, A305844, A306006, A319774, A327052.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Aug 18 2019
%E a(5)-a(7) from _Christian Sievers_, Oct 22 2023
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