Ordered cycle lengths in a random permutation

Abstract

Let x(t) denote the number of jumps occurring in the time interval [0, t) and vk(t) = P(x(t) = k). The generating function of Vk(t) is given by exp (FORMULA PRESENTGED). Lay off to the right of the origin successive intervals of length zjjα, j = 1, 2,& Explicitly the end points are t1 = 0 (FORMULA PRESENTGED). Following Shepp and Lloyd Lr, the length of the rth longest cycle and Sr 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 Lr and Sr 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. © 1971 Pacific Journal Of Mathematics.