cp's OEIS Frontend

A327039 Number of set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

%I A327039 #8 Oct 22 2023 16:49:13
%S A327039 1,2,7,88,25421,2077323118,9221293242272922067,
%T A327039 170141182628636920942528022609657505092
%N A327039 Number of set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).
%C A327039 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 A327039 Binomial transform of A327040.
%e A327039 The a(0) = 1 through a(2) = 7 set-systems:
%e A327039   {}  {}     {}
%e A327039       {{1}}  {{1}}
%e A327039              {{2}}
%e A327039              {{1,2}}
%e A327039              {{1},{1,2}}
%e A327039              {{2},{1,2}}
%e A327039              {{1},{2},{1,2}}
%t A327039 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A327039 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A327039 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]
%Y A327039 The unlabeled multiset partition version is A319752.
%Y A327039 The BII-numbers of these set-systems are A326853.
%Y A327039 The pairwise intersecting case is A327038.
%Y A327039 The covering case is A327040.
%Y A327039 The case where the dual is strict is A327052.
%Y A327039 Cf. A058891, A306006, A319765, A327020, A327053.
%K A327039 nonn,more
%O A327039 0,2
%A A327039 _Gus Wiseman_, Aug 17 2019
%E A327039 a(5)-a(7) from _Christian Sievers_, Oct 22 2023