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A365534 Number of convergent Boolean relation matrices on [n]. 8
1, 2, 15, 465, 61068, 32453533, 67904955351 (list; graph; refs; listen; history; text; internal format)
A Boolean relation matrix R is convergent iff R^k = R^(k+1) for all sufficiently large k. In other words, iff the period of R is equal to 1. The digraph of R is such that all its maximal cyclic nets are primitive (A070322) iff R is convergent. Cf. Rosenblatt link. Also, R is convergent iff every diagonal block in its Frobenius normal form is either primitive or a 1 X 1 zero matrix, Theorem 1.1 in Gregory, Kirkland and Pullman.
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.
G. Markowsky, Bounds on the index and period of a binary relation on a finite set, Semigroup Forum, Vol 13 (1977), 253-259.
E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.
D. Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.
Sum_{n>=0} a_n*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(-(p(x)-1+x))) where E(x) = Sum_{n>=0} x^n/(n!*2^binomial(n,2)), p(x) = Sum_{n>=0} A070322(n)x^n/n! and @ is the exponential Hadamard product (see Panafieu and Dovgal).
A070322(n) <= a(n) <= 2^(n^2) = A002416(n).
Sequence in context: A363834 A013059 A013098 * A158109 A357856 A177394
Geoffrey Critzer, Sep 08 2023

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Last modified December 9 01:07 EST 2023. Contains 367681 sequences. (Running on oeis4.)