A059203 -id:A059203 - cp's OEIS frontend

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

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

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 16 2001

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)

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

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.

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 *)

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

A059202 Triangle T(n,m) of numbers of m-block T_0-covers of a labeled n-set, m = 0..2^n - 1.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 1, 0, 0, 3, 29, 35, 21, 7, 1, 0, 0, 0, 140, 1015, 2793, 4935, 6425, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 0, 0, 0, 420, 13965, 126651, 661801, 2533135, 7792200, 20085000, 44307120, 84651840, 141113700, 206251500, 265182300

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 18 2001

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.

Also, T(n,m) is the number of n X m (0,1)-matrices with pairwise distinct nonzero columns and pairwise distinct nonzero rows, up to permutation of columns.

			[1],
[0,1],
[0,0,3,1],
[0,0,3,29,35,21,7,1],
...
There are 35 4-block T_0-covers of a labeled 3-set.

Robert Israel, Table of n, a(n) for n = 0..4094 (rows 0 to 11, flattened).
T_0-covers of a labeled 3-set

Cf. A059201 (row sums), A059203 (column sums), A094000 (main diagonal).
Cf. A059084, A059085, A059086, A059087, A059088, A059089, A181230.
Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

with(combinat): for n from 0 to 10 do for m from 0 to 2^n-1 do printf(`%d,`,(1/m!)*sum(stirling1(m+1,i)*product(2^(i-1)-1-j, j=0..n-1), i=1..m+1)) od: od:

T[n_, m_] = Sum[ StirlingS1[n + 1, i + 1]*Binomial[2^i - 1, m], {i, 0, n}]; Table[T[n, m], {n, 0, 5}, {m, 0, 2^n - 1}] (* G. C. Greubel, Dec 28 2016 *)

T(n, m) = (1/m!)*Sum_{1..m + 1} stirling1(m + 1, i)*[2^(i - 1) - 1]_n, where [k]_n := k*(k - 1)*...*(k - n + 1), [k]_0 = 1.
E.g.f: Sum((1+x)^(2^n-1)*log(1+y)^n/n!, n=0..infinity)/(1+y). - Vladeta Jovovic, May 19 2004
Also T(n, m) = Sum_{i=0..n} Stirling1(n+1, i+1)*binomial(2^i-1, m). - Vladeta Jovovic, Jun 04 2004
T(n,m) = A181230(n,m)/m! - n*T(n-1,m) - T(n,m-1) - n*T(n-1,m-1). - Max Alekseyev, Dec 11 2017

More terms from James Sellers, Jan 24 2001

cp's OEIS Frontend

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

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