A327039 - OEIS

%I #8 Oct 22 2023 16:49:13

%S 1,2,7,88,25421,2077323118,9221293242272922067,

%T 170141182628636920942528022609657505092

%N Number of 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 that are cointersecting, meaning their dual is pairwise intersecting.

%F Binomial transform of A327040.

%e The a(0) = 1 through a(2) = 7 set-systems:

%e   {}  {}     {}

%e       {{1}}  {{1}}

%e              {{2}}

%e              {{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}]],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 pairwise intersecting case is A327038.

%Y The covering case is A327040.

%Y The case where the dual is strict is A327052.

%Y Cf. A058891, A306006, A319765, A327020, A327053.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Aug 17 2019

%E a(5)-a(7) from _Christian Sievers_, Oct 22 2023