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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 707.11071

**Autor: ** Erdös, Paul; Szalay, M.

**Title: ** On some problems of the statistical theory of partitions. (In English)

**Source: ** Number theory. Vol. I. Elementary and analytic, Proc. Conf., Budapest/Hung. 1987, Colloq. Math. Soc. János Bolyai 51, 93-110 (1990).

**Review: ** [For the entire collection see Zbl 694.00005.]

Let \pi be a generic ``unrestricted'' partition of the positive integer n, that is, a partition \lambda_{1}+\lambda_{2}+...+\lambda_{m} = n, where the \lambda_{j}'s are integers such that \lambda_{1} \geq \lambda_{2} \geq ... \geq \lambda_{m}, and let p(n) be the number of such partitions. The number of conjugacy classes of the symmetric group of degree n is equal to p(n), and the number of conjugacy classes of the alternating group of degree n is asymptotically equal to p(n)/2. By choosing a suitable prime summand, a proof that almost all partitions \pi of n have a summand which is > 1 and relatively prime to the other summands was given by *L. B. Beasley*, *J. L. Brenner*, *P. Erdös*, *M. Szalay* and *A. G. Williamson* [Period. Math. Hung. 18, 259-269 (1987; Zbl 617.20045)], and was used to simplify a proof originally given by Beasley, Brenner, and Williamson that almost all conjugacy classes of the alternating group of degree n contain a pair of generators.

It is now shown that the choice of a prime summand was necessary, in the sense that for almost all \pi's, if \lambda_{j} > 1 and (\lambda_{i},\lambda_{j}) = 1 for each i\ne j then \lambda_{j} is a prime. Also, let \pi^{x} be a generic unequal partition of n, that is, \pi^{x} represents a partition \alpha_{1}+\alpha_{2}+...+\alpha_{m} = n, where the \alpha_{j}'s are integers such that \alpha_{1} > \alpha_{2} > ... > \alpha_{m}, and let M(\pi^{x}) denote the maximal number of consecutive summands in \pi^{x}. It is shown that for almost all \pi^{x}, M(\pi^{x}) = (log n)/(2 log 2)-(log log n)/(log 2)+O(\omega(n)), where \omega(n) ––> oo (arbitrarily slowly). Finally, let T_{n}(k) denote the number of solutions of x^{k} = e in the symmetric group of degree n, where e is the identity element. Others have investigated the behavior of T_{n}(k) as n ––> oo for fixed k \geq 2. An estimate is now established for T_{n}(k) for 1 \leq k \leq n^{(1/4)-\epsilon}, 0 < \epsilon < 10^{-2}, as n ––> oo.

**Reviewer: ** B.Garrison

**Classif.: ** * 11P81 Elementary theory of partitions

20P05 Probability methods in group theory

00A07 Problem books

**Keywords: ** unequal partition

**Citations: ** Zbl 626.20059; Zbl 694.00005; Zbl 617.20045

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