A327053 Number of T_0 (costrict) set-systems covering n vertices where every two vertices appear together in some edge (cointersecting).

1, 1, 3, 62, 24710, 2076948136, 9221293198653529144, 170141182628636920684331812494864430896

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 whose dual is strict and pairwise intersecting.

Examples

			The a(1) = 1 through a(2) = 3 set-systems:
  {}  {{1}}  {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}
The a(3) = 62 set-systems:
  1 2 123    1 2 3 123    1 2 12 13 23   1 2 3 12 13 23   1 2 3 12 13 23 123
  1 3 123    1 12 13 23   1 2 3 12 123   1 2 3 12 13 123
  2 3 123    1 2 12 123   1 2 3 13 123   1 2 3 12 23 123
  1 12 123   1 2 13 123   1 2 3 23 123   1 2 3 13 23 123
  1 13 123   1 2 23 123   1 3 12 13 23   1 2 12 13 23 123
  12 13 23   1 3 12 123   2 3 12 13 23   1 3 12 13 23 123
  2 12 123   1 3 13 123   1 2 12 13 123  2 3 12 13 23 123
  2 23 123   1 3 23 123   1 2 12 23 123
  3 13 123   2 12 13 23   1 2 13 23 123
  3 23 123   2 3 12 123   1 3 12 13 123
  12 13 123  2 3 13 123   1 3 12 23 123
  12 23 123  2 3 23 123   1 3 13 23 123
  13 23 123  3 12 13 23   2 3 12 13 123
             1 12 13 123  2 3 12 23 123
             1 12 23 123  2 3 13 23 123
             1 13 23 123  1 12 13 23 123
             2 12 13 123  2 12 13 23 123
             2 12 23 123  3 12 13 23 123
             2 13 23 123
             3 12 13 123
             3 12 23 123
             3 13 23 123
             12 13 23 123
		

Crossrefs

The pairwise intersecting case is A319774.

The BII-numbers of these set-systems are the intersection of A326947 and A326853.

The non-T_0 version is A327040.

The non-covering version is A327052.

Cf. A003465, A305843, A319767, A326854, A327020, A327037, A327039.

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

Formula

Inverse binomial transform of A327052.

Extensions

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