A327037 Number of pairwise intersecting set-systems covering n vertices where every two vertices appear together in some edge (cointersecting).

1, 1, 3, 21, 913, 1183295, 909142733955, 291200434282476769116160

Offset: 0

Author

Gus Wiseman, Aug 17 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 pairwise intersecting, covering set-systems that are cointersecting, meaning their dual is pairwise intersecting.

Examples

			The a(0) = 1 through a(3) = 21 set-systems:
  {}  {{1}}  {{1,2}}      {{1,2,3}}
             {{1},{1,2}}  {{1},{1,2,3}}
             {{2},{1,2}}  {{2},{1,2,3}}
                          {{3},{1,2,3}}
                          {{1,2},{1,2,3}}
                          {{1,3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{1,2},{1,3},{2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1,2},{1,3},{1,2,3}}
                          {{1,2},{2,3},{1,2,3}}
                          {{1,3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}
                          {{1,2},{1,3},{2,3},{1,2,3}}
		

Crossrefs

Intersecting covering set-systems are A305843.

The unlabeled multiset partition version is A319765.

The case where the dual is strict is A319774.

The BII-numbers of these set-systems are A326912.

The non-covering version is A327038.

Cointersectng covering set-systems are A327040.

Cf. A003465, A319767, A326853, A326854, A326910.

Programs

Mathematica

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],Union@@#==Range[n]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]

Formula

Inverse binomial transform of A327038.

Extensions

a(6)-a(7) from Christian Sievers, Aug 18 2024