cp's OEIS Frontend

A327040 Number of set-systems covering n vertices, every two of which appear together in some edge (cointersecting).

%I A327040 #9 Oct 22 2023 16:49:34
%S A327040 1,1,4,72,25104,2077196832,9221293229809363008,
%T A327040 170141182628636920877978969957369949312
%N A327040 Number of set-systems covering n vertices, every two of which appear together in some edge (cointersecting).
%C A327040 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 that are cointersecting, meaning their dual is pairwise intersecting.
%F A327040 Inverse binomial transform of A327039.
%e A327040 The a(0) = 1 through a(2) = 4 set-systems:
%e A327040   {}  {{1}}  {{1,2}}
%e A327040              {{1},{1,2}}
%e A327040              {{2},{1,2}}
%e A327040              {{1},{2},{1,2}}
%t A327040 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A327040 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A327040 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]
%Y A327040 The unlabeled multiset partition version is A319752.
%Y A327040 The BII-numbers of these set-systems are A326853.
%Y A327040 The antichain case is A327020.
%Y A327040 The pairwise intersecting case is A327037.
%Y A327040 The non-covering version is A327039.
%Y A327040 The case where the dual is strict is A327053.
%Y A327040 Cf. A003465, A305843, A305844, A306006, A319774, A327052.
%K A327040 nonn,more
%O A327040 0,3
%A A327040 _Gus Wiseman_, Aug 18 2019
%E A327040 a(5)-a(7) from _Christian Sievers_, Oct 22 2023