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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. 1

%I #14 Sep 30 2023 21:46:34

%S 1,1,1,3,7,3,25,85,84,25,543,2335,3579,2322,543,29281,152101,310020,

%T 309725,151835,29281,3781503,23139487,58538763,78349050,58514700,

%U 23128233,3781503,1138779265,8051910805,24318772884,40667112045,40664902810,24315521720,8050866418,1138779265

%N 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 A 1(0) block is such that every entry in the block is 1(0).

%C Conjecture: lim_{n -> oo} T(n,k)/T(n,n-k) = 1.

%H 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 T(n,0) = T(n,n) = A003024(n).

%F E.g.f.: D(y(exp(x)-1)+x) where D(x) is the e.g.f. for A003024.

%e Triangle begins ...

%e 1;

%e 1, 1;

%e 3, 7, 3;

%e 25, 85, 84, 25;

%e 543, 2335, 3579, 2322, 543;

%e 29281, 152101, 310020, 309725, 151835, 29281;

%e 3781503, 23139487, 58538763, 78349050, 58514700, 23128233, 3781503;

%e ...

%t nn = 6; B[n_] := 2^Binomial[n, 2] n!; dags=Select[Import["https://oeis.org/A003024/b003024.txt", "Table"],

%t Length@# == 2 &][[All, 2]]; d[x_] := Total[dags Table[x^i/i!, {i, 0, 40}]];

%t Map[Select[#, # > 0 &] &,Table[n!, {n, 0, nn}] CoefficientList[

%t Series[d[y (Exp[x] - 1) + x], {x, 0, nn}], {x, y}]] // Grid

%Y Cf. A365593 (row sums), A003024.

%K nonn,tabl

%O 0,4

%A _Geoffrey Critzer_, Sep 30 2023

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Last modified August 11 09:55 EDT 2024. Contains 375059 sequences. (Running on oeis4.)