A366218 Number of convergent binary relations on [n] (A365534) that converge to an equivalence relation (A000110).

1, 1, 4, 149, 26177, 18211032, 47135163595

Offset

0,3

Comments

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.

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.

Links

Table of n, a(n) for n=0..6.

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

D. Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.

Formula

E.g.f.: exp(p(x)-1) where p(x) is the e.g.f. for A070322.

Mathematica

nn = 13; B[n_] := 2^Binomial[n, 2] n!; 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}]]; Table[n!, {n, 0, nn}] CoefficientList[Series[Exp[pr[x] - 1], {x, 0, nn}], x]

Crossrefs

Cf. A070322, A365534, A000110.

Sequence in context: A350985 A269136 A226073 * A361774 A294826 A247115

Adjacent sequences: A366215 A366216 A366217 * A366219 A366220 A366221

Keyword

nonn,more

Author

Geoffrey Critzer, Oct 04 2023

Status

approved