tailieunhanh - Báo cáo toán học: "THE AVERAGE ORDER OF A PERMUTATION"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: THE AVERAGE ORDER OF A PERMUTATION. | THE AVERAGE ORDER OF A PERMUTATION Richard Stong Department of Mathematics Rice University Houston TX 77005 stong@ Submitted May 11 1998 Accepted June 23 1998 Abstract. We show that the average order jin of a permutation in Sn satisfies I n pn loglog n tog fin o o ----- y log n log n J which refines earlier results of Erdos and Turan Schmutz and Goh and Schmutz. 1. Introduction. For ơ 2 Sn let N ơ be the order of ơ in the group Sn. Erdos and Turán 2 showed that if one chooses a permutation uniformly at random from Sn then for n large log N ơ is asymptotically normal with mean log2 n 2 and variance log3 n 3. Define the average order of an element of Sn to be fin n X N ơ . 2Sn It turns out that log fin is much larger than log2 n 2 being dominated by the contribution of a relatively small number of permutations of very high order. This was first shown by Erdos and Turán 3 who showed that log fin O ypn log n . This result was sharpened by Schmutz 6 and later by Goh and Schmutz 4 to show that log fin Cỵ n log n for an explicit constant C. The purpose of this note is to show that log fin o l g og o log n log n where C . is an explicit constant defined below. Our argument shares some similarities with that of 4 but is more elementary and permits a more explicit 1991 Mathematics Subject Classification. Primary 11N37. 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 5 1998 R41 2 bound on the error term. The proof will be divided into three steps. First we will give upper and lower bounds on dn involving the coefficients of a certain power series then we will use a Tauberian theorem to bound these coefficients. For a partition A Al. A2. As let ci A be the number of parts of A of size i let A A1 A2 As and let m A . A1. A2. As . We will say that A is a partition of A . By a sub-partition of A we will mean any subset of A1. A2. As viewed as a partition of some smaller number. Then dn m A 1C1 A 2c2 A . c1 A c2 A . A n 2. The Upper Bound. Call a partition fl