Publications of (and about) Paul Erdös
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 \lambda1+\lambda2+...+\lambdam = n, where the \lambdaj's are integers such that \lambda1 \geq \lambda2 \geq ... \geq \lambdam, 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 \lambdaj > 1 and (\lambdai,\lambdaj) = 1 for each i\ne j then \lambdaj is a prime. Also, let \pix be a generic unequal partition of n, that is, \pix represents a partition \alpha1+\alpha2+...+\alpham = n, where the \alphaj's are integers such that \alpha1 > \alpha2 > ... > \alpham, and let M(\pix) denote the maximal number of consecutive summands in \pix. It is shown that for almost all \pix, M(\pix) = (log n)/(2 log 2)-(log log n)/(log 2)+O(\omega(n)),
where \omega(n) > oo (arbitrarily slowly). Finally, let Tn(k) denote the number of solutions of xk = e in the symmetric group of degree n, where e is the identity element. Others have investigated the behavior of Tn(k) as n > oo for fixed k \geq 2. An estimate is now established for Tn(k) for 1 \leq k \leq n(1/4)-\epsilon, 0 < \epsilon < 10-2, as n > oo.
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
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag