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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)
 OFFSET 0,2 COMMENTS 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. 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, 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. FORMULA 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). CROSSREFS Cf. A070322, A002416. Sequence in context: A363834 A013059 A013098 * A158109 A357856 A177394 Adjacent sequences: A365531 A365532 A365533 * A365535 A365536 A365537 KEYWORD nonn,more AUTHOR Geoffrey Critzer, Sep 08 2023 STATUS approved

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