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%I #14 Oct 05 2023 08:38:04
%S 1,1,4,149,26177,18211032,47135163595
%N Number of convergent binary relations on [n] (A365534) that converge to an equivalence relation (A000110).
%C Equivalently, a(n) is the number of Boolean relation matrices whose Frobenius normal form is such that all the diagonal blocks are primitive (A070322) and all the off diagonal blocks are 0-blocks. See Gregory, Kirkland, Pullman.
%C The limit of a convergent binary relation R is an equivalence relation iff every vertex and every edge in G(R) is on a cycle, where G(R) is the directed graph with loops associated to R. See Corollary to Theorem 1 in Rosenblatt.
%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, pp. 105-117.
%H 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 E.g.f.: exp(p(x)-1) where p(x) is the e.g.f. for A070322.
%t nn = 13; B[n_] := 2^Binomial[n, 2] n!; primitive = Select[Import["https://oeis.org/A070322/b070322.txt", "Table"],
%t Length@# == 2 &][[All, 2]];pr[x_] := Total[primitive Table[x^i/i!, {i, 0, 6}]];Table[n!, {n, 0, nn}] CoefficientList[Series[Exp[pr[x] - 1], {x, 0, nn}], x]
%Y Cf. A070322, A365534, A000110.
%K nonn,more
%O 0,3
%A _Geoffrey Critzer_, Oct 04 2023
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