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%I #9 Feb 04 2024 12:39:01
%S 1,2,6,75,24981,2077072342,9221293211115589902,
%T 170141182628636920748880864929055912851
%N Number of T_0 (costrict) 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 set-systems whose dual is strict and pairwise intersecting.
%F Binomial transform of A327053.
%e The a(0) = 1 through a(2) = 6 set-systems:
%e {} {} {}
%e {{1}} {{1}}
%e {{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}]],UnsameQ@@dual[#]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]
%Y The unlabeled multiset partition version is A319760.
%Y The non-T_0 version is A327039.
%Y The covering case is A327053.
%Y Cf. A051185, A319752, A319774, A327038, A327040.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Aug 18 2019
%E a(5)-a(7) from _Christian Sievers_, Feb 04 2024
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