A327052 Number of T_0 (costrict) set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

1, 2, 6, 75, 24981, 2077072342, 9221293211115589902, 170141182628636920748880864929055912851

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 set-systems whose dual is strict and pairwise intersecting.

Examples

			The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}
		

Crossrefs

The unlabeled multiset partition version is A319760.

The non-T_0 version is A327039.

The covering case is A327053.

Cf. A051185, A319752, A319774, A327038, A327040.

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}]],UnsameQ@@dual[#]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]

Formula

Binomial transform of A327053.

Extensions

a(5)-a(7) from Christian Sievers, Feb 04 2024