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

%I A059202 #28 Jul 02 2025 16:02:00
%S A059202 1,0,1,0,0,3,1,0,0,3,29,35,21,7,1,0,0,0,140,1015,2793,4935,6425,6435,
%T A059202 5005,3003,1365,455,105,15,1,0,0,0,420,13965,126651,661801,2533135,
%U A059202 7792200,20085000,44307120,84651840,141113700,206251500,265182300
%N A059202 Triangle T(n,m) of numbers of m-block T_0-covers of a labeled n-set, m = 0..2^n - 1.
%C A059202 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.
%C A059202 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.
%H A059202 Robert Israel, <a href="/A059202/b059202.txt">Table of n, a(n) for n = 0..4094</a> (rows 0 to 11, flattened).
%H A059202 <a href="/A059201/a059201.pdf">T_0-covers of a labeled 3-set</a>
%F A059202 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.
%F A059202 E.g.f: Sum((1+x)^(2^n-1)*log(1+y)^n/n!, n=0..infinity)/(1+y). - _Vladeta Jovovic_, May 19 2004
%F A059202 Also T(n, m) = Sum_{i=0..n} Stirling1(n+1, i+1)*binomial(2^i-1, m). - _Vladeta Jovovic_, Jun 04 2004
%F A059202 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
%e A059202 [1],
%e A059202 [0,1],
%e A059202 [0,0,3,1],
%e A059202 [0,0,3,29,35,21,7,1],
%e A059202 ...
%e A059202 There are 35 4-block T_0-covers of a labeled 3-set.
%p A059202 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 A059202 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 *)
%Y A059202 Cf. A059201 (row sums), A059203 (column sums), A094000 (main diagonal).
%Y A059202 Cf. A059084, A059085, A059086, A059087, A059088, A059089, A181230.
%Y A059202 Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763
%K A059202 easy,nonn,tabf
%O A059202 0,6
%A A059202 _Vladeta Jovovic_, Goran Kilibarda, Jan 18 2001
%E A059202 More terms from _James Sellers_, Jan 24 2001