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A366141 Triangular array read by rows: T(n,k) is the number of Boolean relation matrices such that all of the blocks of its Frobenius normal form are 0-blocks or 1-blocks and that have exactly k 1-blocks on the diagonal, n>=0, 0<=k<=n.

%I A366141 #14 Sep 30 2023 21:46:34
%S A366141 1,1,1,3,7,3,25,85,84,25,543,2335,3579,2322,543,29281,152101,310020,
%T A366141 309725,151835,29281,3781503,23139487,58538763,78349050,58514700,
%U A366141 23128233,3781503,1138779265,8051910805,24318772884,40667112045,40664902810,24315521720,8050866418,1138779265
%N A366141 Triangular array read by rows: T(n,k) is the number of Boolean relation matrices such that all of the blocks of its Frobenius normal form are 0-blocks or 1-blocks and that have exactly k 1-blocks on the diagonal, n>=0, 0<=k<=n.
%C A366141 A 1(0) block is such that every entry in the block is 1(0).
%C A366141 Conjecture:  lim_{n -> oo} T(n,k)/T(n,n-k) = 1.
%H A366141 D. A. Gregory, S. Kirkland, and N. J. Pullman, <a href="https://doi.org/10.1016/0024-3795(93)90323-G">Power convergent Boolean matrices</a>, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.
%F A366141 T(n,0) = T(n,n) = A003024(n).
%F A366141 E.g.f.: D(y(exp(x)-1)+x) where D(x) is the e.g.f. for A003024.
%e A366141 Triangle begins ...
%e A366141         1;
%e A366141         1,        1;
%e A366141         3,        7,        3;
%e A366141        25,       85,       84,       25;
%e A366141       543,     2335,     3579,     2322,      543;
%e A366141     29281,   152101,   310020,   309725,   151835,    29281;
%e A366141   3781503, 23139487, 58538763, 78349050, 58514700, 23128233, 3781503;
%e A366141   ...
%t A366141 nn = 6; B[n_] := 2^Binomial[n, 2] n!; dags=Select[Import["https://oeis.org/A003024/b003024.txt", "Table"],
%t A366141    Length@# == 2 &][[All, 2]]; d[x_] := Total[dags Table[x^i/i!, {i, 0, 40}]];
%t A366141 Map[Select[#, # > 0 &] &,Table[n!, {n, 0, nn}] CoefficientList[
%t A366141     Series[d[y (Exp[x] - 1) + x], {x, 0, nn}], {x, y}]] // Grid
%Y A366141 Cf. A365593 (row sums), A003024.
%K A366141 nonn,tabl
%O A366141 0,4
%A A366141 _Geoffrey Critzer_, Sep 30 2023