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%I #9 Feb 04 2024 12:39:31
%S 1,1,3,62,24710,2076948136,9221293198653529144,
%T 170141182628636920684331812494864430896
%N Number of T_0 (costrict) 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 covering set-systems whose dual is strict and pairwise intersecting.
%F Inverse binomial transform of A327052.
%e The a(1) = 1 through a(2) = 3 set-systems:
%e {} {{1}} {{1},{1,2}}
%e {{2},{1,2}}
%e {{1},{2},{1,2}}
%e The a(3) = 62 set-systems:
%e 1 2 123 1 2 3 123 1 2 12 13 23 1 2 3 12 13 23 1 2 3 12 13 23 123
%e 1 3 123 1 12 13 23 1 2 3 12 123 1 2 3 12 13 123
%e 2 3 123 1 2 12 123 1 2 3 13 123 1 2 3 12 23 123
%e 1 12 123 1 2 13 123 1 2 3 23 123 1 2 3 13 23 123
%e 1 13 123 1 2 23 123 1 3 12 13 23 1 2 12 13 23 123
%e 12 13 23 1 3 12 123 2 3 12 13 23 1 3 12 13 23 123
%e 2 12 123 1 3 13 123 1 2 12 13 123 2 3 12 13 23 123
%e 2 23 123 1 3 23 123 1 2 12 23 123
%e 3 13 123 2 12 13 23 1 2 13 23 123
%e 3 23 123 2 3 12 123 1 3 12 13 123
%e 12 13 123 2 3 13 123 1 3 12 23 123
%e 12 23 123 2 3 23 123 1 3 13 23 123
%e 13 23 123 3 12 13 23 2 3 12 13 123
%e 1 12 13 123 2 3 12 23 123
%e 1 12 23 123 2 3 13 23 123
%e 1 13 23 123 1 12 13 23 123
%e 2 12 13 123 2 12 13 23 123
%e 2 12 23 123 3 12 13 23 123
%e 2 13 23 123
%e 3 12 13 123
%e 3 12 23 123
%e 3 13 23 123
%e 12 13 23 123
%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]&&UnsameQ@@dual[#]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]
%Y The pairwise intersecting case is A319774.
%Y The BII-numbers of these set-systems are the intersection of A326947 and A326853.
%Y The non-T_0 version is A327040.
%Y The non-covering version is A327052.
%Y Cf. A003465, A305843, A319767, A326854, A327020, A327037, A327039.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Aug 18 2019
%E a(5)-a(7) from _Christian Sievers_, Feb 04 2024
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