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, 1, 1, 3, 7, 3, 25, 85, 84, 25, 543, 2335, 3579, 2322, 543, 29281, 152101, 310020, 309725, 151835, 29281, 3781503, 23139487, 58538763, 78349050, 58514700, 23128233, 3781503, 1138779265, 8051910805, 24318772884, 40667112045, 40664902810, 24315521720, 8050866418, 1138779265

Offset

0,4

Comments

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

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

Links

Table of n, a(n) for n=0..35.

D. A. Gregory, S. Kirkland, and N. J. Pullman, Power convergent Boolean matrices, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.

Formula

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

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

Example

Triangle begins ...
        1;
        1,        1;
        3,        7,        3;
       25,       85,       84,       25;
      543,     2335,     3579,     2322,      543;
    29281,   152101,   310020,   309725,   151835,    29281;
  3781503, 23139487, 58538763, 78349050, 58514700, 23128233, 3781503;
  ...

Mathematica

nn = 6; B[n_] := 2^Binomial[n, 2] n!; dags=Select[Import["https://oeis.org/A003024/b003024.txt", "Table"],
   Length@# == 2 &][[All, 2]]; d[x_] := Total[dags Table[x^i/i!, {i, 0, 40}]];
Map[Select[#, # > 0 &] &, Table[n!, {n, 0, nn}] CoefficientList[
    Series[d[y (Exp[x] - 1) + x], {x, 0, nn}], {x, y}]] // Grid

Crossrefs

Cf. A365593 (row sums), A003024.

Sequence in context: A019158 A367865 A086153 * A049479 A125314 A213244

Adjacent sequences: A366138 A366139 A366140 * A366142 A366143 A366144

Keyword

nonn,tabl

Author

Geoffrey Critzer, Sep 30 2023

Status

approved