The Expected Order of a Random Permutation

Let fin be the expected order of a random permutation, that is, the arithmetic mean of the orders of the elements in the symmetric group Sn. We prove that log/in ~ c\/(n/\ogn) as n -> oo, where c = 2 / ( 2 I °° log log I — — I dt\ 1. Overview If a is a permutation on n letters, let Nn{a) be the order of a as a group element. For a typical permutation, Nn is about n . To make this precise, we quote a stronger result of Erdos and Turan [5]. THEOREM 1. For any fixed x, # j cr e 5 n : log (A^n(a)) < | log « + -^log 3 / 2 « | ^ n-»oo v \^"'^J J — oo Many authors have given their own proofs of this remarkable theorem. For a survey of these and related results, see [2]. Let un = —V (order of a) n\ ^^ be the expected order of a random permutation. The problem of estimating fin was first raised by Erdos and Turan [6]. Note that x is fixed in Theorem 1. It will not help us estimate //n because we cannot ignore the tail of the distribution. There are some permutations for which Nn is quite large. In fact, Landau proved that ma.\Nn(a) = e . It turns out that a small set of exceptional permutations contributes significantly to fj,n. Erdos and Turan determined that log//n = O(\/(n/\og «)). (A proof appears in [12, 13].) This paper contains sharper estimates. We prove the following asymptotic formula. Received 25 August 1989; revised 1 July 1990. 1980 Mathematics Subject Classification 11N37. Research supported by NSF (DMS-8901610) and a Drexel University Faculty Development Minigrant. Bull. London Math. Soc. 23 (1991) 34-42 THE EXPECTED ORDER OF A RANDOM PERMUTATION 35 THEOREM 2. c 2 7( 2 f First we give a brief overview of the proof. Consider the generating function ZBnent = (\-e-r U (1 log (1-«"*)). n primes p One can think of Bn as the sum of the weights of a certain set of weighted partitions. By classical methods, one can easily prove that logi?n ~ c\/(n/\ogn). Our goal is to prove that \og/in ~ logl?n. The connection between permutations and partitions is that the cycle lengths of a permutation on n letters form a partition X of the integer n (written X I— «). By a wellknown theorem of Cauchy, the number of permutations of n letters with ct cycles of length / is n\ Cl\c2\...cn\\ i2*...nn' If X = {XvX2,...} = {\\2\...,n »}, define LCM(Al5/l2)...) w(xy.= Since the order of a permutation is the least common multiple of its cycle lengths, we have Hn = E X\-n For each fixed integer s ̂ 2, we shall construct a set P^ of partitions of n. Obviously, a lower bound can be obtained by considering only the contribution from elements of/*: The idea is to choose P°n in such a way that the right-hand side of (•) is both easy to estimate and large enough to give a good bound. Consider the generating function ( p-2pt p-3pt p-ipt f,spt\ l+e-» + —+ -— + —+... + — . j / We shall choose P^ in such a way that For large s, this is a good approximation; for any £ > 0, one can choose s so that, for n > no(e, s), one has