Ordered cycle lengths in a random permutation

Abstract

Let x(t) denote the number of jumps occurring in the time interval [0, t) and v k (t) = P{x(f) = k}. The generating function of Vk(t) is given by exp {t[(x) -1]}, (x) = P*&f P* = 1 fc = l fc = l Lay off to the right of the origin successive intervals of length &H a $ 3 = 1 2, . Explicitly the end points are 1 = 0 = */*", 3 = 2, 3, , a > 0 , and a Following Shepp and Lloyd L r , the length of the rth longest cycle and S r , the length of the rth shortest cycle have been defined for our choice of x(t) and tj, j = 1,2, . This paper obtains the asymptotics for the mth moments of L r and S r suitably normalized by a new technique of generating functions. It is further shown that the results of Shepp and Lloyd are particular cases of these more general results.