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A365547 Triangular array read by rows. T(n,k) is the number of convergent Boolean relation matrices on [n] containing exactly k strongly connected components, n>=0, 0<=k<=n.

%I A365547 #13 Sep 11 2023 11:24:53
%S A365547 1,0,2,0,3,12,0,139,126,200,0,25575,17517,9288,8688,0,18077431,
%T A365547 8457840,3545350,1435920,936992,0,47024942643,14452288791,4277647665,
%U A365547 1422744780,485315280,242016192
%N A365547 Triangular array read by rows.  T(n,k) is the number of convergent Boolean relation matrices on [n] containing exactly k strongly connected components, n>=0, 0<=k<=n.
%H A365547 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.
%H A365547 G. Markowsky, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN001251775">Bounds on the index and period of a binary relation on a finite set</a>, Semigroup Forum, Vol 13 (1977), 253-259.
%H A365547 E. de Panafieu and S. Dovgal, <a href="https://arxiv.org/abs/1903.09454">Symbolic method and directed graph enumeration</a>, arXiv:1903.09454 [math.CO], 2019.
%H A365547 R. W. Robinson, <a href="http://cobweb.cs.uga.edu/~rwr/publications/components.pdf">Counting digraphs with restrictions on the strong components</a>, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.
%H A365547 D. Rosenblatt, <a href="https://nvlpubs.nist.gov/nistpubs/jres/67B/jresv67Bn4p249_A1b.pdf">On the graphs of finite Boolean relation matrices</a>, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.
%F A365547  For n>=2, T(n,1) = A070322(n) and T(n,n) = A003024(n)*2^n.
%e A365547  Triangle begins ...
%e A365547   1;
%e A365547   0,        2;
%e A365547   0,        3,      12;
%e A365547   0,      139,     126,     200;
%e A365547   0,    25575,   17517,    9288,    8688;
%e A365547   0, 18077431, 8457840, 3545350, 1435920, 936992;
%e A365547   ...
%t A365547 nn = 6; B[n_] := n! 2^Binomial[n, 2]; primitive = Select[Import["https://oeis.org/A070322/b070322.txt", "Table"], Length@# == 2 &][[All, 2]]; pr[x_] := Total[primitive Table[x^i/i!, {i, 0, 6}]];
%t A365547 ggf[egf_] := Normal[Series[egf, {x, 0, nn}]] /. Table[x^i -> x^i/2^Binomial[i, 2], {i, 0, nn}];Table[Take[(Table[B[n], {n, 0, nn}] CoefficientList[Series[1/ggf[Exp[-(y pr[x] - y + y x)]], {x, 0, nn}], {x, y}])[[i]], i], {i, 1, 7}] // Grid
%Y A365547 Cf. A365534 (row sums), A070322, A003024.
%K A365547 nonn,tabl
%O A365547 0,3
%A A365547 _Geoffrey Critzer_, Sep 08 2023