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%I #6 Aug 18 2019 11:27:40
%S 1,2,6,34,1020,1188106
%N Number of pairwise intersecting set-systems covering a subset of {1..n} where every two covered 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 set-systems that are cointersecting, meaning their dual is pairwise intersecting.
%F Binomial transform of A327037.
%e The a(0) = 1 through a(2) = 6 set-systems:
%e {} {} {}
%e {{1}} {{1}}
%e {{2}}
%e {{1,2}}
%e {{1},{1,2}}
%e {{2},{1,2}}
%e The a(3) = 34 set-systems:
%e {} {{1}} {{1}{12}} {{1}{12}{123}} {{1}{12}{13}{123}}
%e {{2}} {{1}{13}} {{1}{13}{123}} {{2}{12}{23}{123}}
%e {{3}} {{2}{12}} {{12}{13}{23}} {{3}{13}{23}{123}}
%e {{12}} {{2}{23}} {{2}{12}{123}} {{12}{13}{23}{123}}
%e {{13}} {{3}{13}} {{2}{23}{123}}
%e {{23}} {{3}{23}} {{3}{13}{123}}
%e {{123}} {{1}{123}} {{3}{23}{123}}
%e {{2}{123}} {{12}{13}{123}}
%e {{3}{123}} {{12}{23}{123}}
%e {{12}{123}} {{13}{23}{123}}
%e {{13}{123}}
%e {{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]=={}&],stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]
%Y Intersecting set-systems are A051185.
%Y The unlabeled multiset partition version is A319765.
%Y The BII-numbers of these set-systems are A326912.
%Y The covering case is A327037.
%Y Cointersecting set-systems are A327039.
%Y The case where the dual is strict is A327040.
%Y Cf. A058891, A319767, A319774, A326854, A327052.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Aug 17 2019
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