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

1, 1, 4, 72, 25104, 2077196832, 9221293229809363008, 170141182628636920877978969957369949312

Offset: 0

Author

Gus Wiseman, Aug 18 2019

Keywords

• nonn
• more

Comments

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.

Examples

			The a(0) = 1 through a(2) = 4 set-systems:
  {}  {{1}}  {{1,2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}
		

Crossrefs

The unlabeled multiset partition version is A319752.

The BII-numbers of these set-systems are A326853.

The antichain case is A327020.

The pairwise intersecting case is A327037.

The non-covering version is A327039.

The case where the dual is strict is A327053.

Cf. A003465, A305843, A305844, A306006, A319774, A327052.

Programs

Mathematica

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]

Formula

Inverse binomial transform of A327039.

Extensions

a(5)-a(7) from Christian Sievers, Oct 22 2023