A059201 Number of T_0-covers of a labeled n-set.

1, 1, 4, 96, 31692, 2147001636, 9223371991763269704, 170141183460469231473432887375376674952, 57896044618658097711785492504343953920509909728243389682424010192567186540224

Offset: 0

Author

Vladeta Jovovic, Goran Kilibarda, Jan 16 2001

Keywords

• easy
• nonn

Comments

A cover of a set is a T_0-cover if for every two distinct points of the set there exists a member (block) of the cover containing one but not the other point.

From Gus Wiseman, Aug 13 2019: (Start)

A set-system is a finite set of finite nonempty sets. 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}}. The T_0 condition means that the dual is strict (no repeated edges). For example, the a(2) = 4 covers are:

{{1},{2}}

{{1},{1,2}}

{{2},{1,2}}

{{1},{2},{1,2}}

(End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..11
• Vladeta Jovovic, T_0-covers of a labeled 3-set

Crossrefs

Row sums of A059202.

Cf. A059203, A059084, A059085, A059086, A059087, A059088, A059089.

Covering set-systems are A003465.

The unlabeled version is A319637.

The version with empty edges allowed is A326939.

The non-covering version is A326940.

BII-numbers of T_0 set-systems are A326947.

The same with connected instead of covering is A326948.

The T_1 version is A326961.

Cf. A245567, A316978, A319559, A319564, A323818, A326941, A326946, A326970.

Programs

Mathematica

Table[Sum[StirlingS1[n + 1, k]*2^(2^(k - 1) - 1), {k, 0, n + 1}], {n,0,5}] (* G. C. Greubel, Dec 28 2016 *)
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&]],{n,0,3}] (* Gus Wiseman, Aug 13 2019 *)

Formula

a(n) = Sum_{i=0..n+1} stirling1(n+1, i)*2^(2^(i-1)-1).

a(n) = Sum_{m=0..2^n-1} A059202(n,m).

Inverse binomial transform of A326940 and exponential transform of A326948. - Gus Wiseman, Aug 13 2019