Vol. 36, No. 3, 1971

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ISSN: 0030-8730
Ordered cycle lengths in a random permutation

V. Balakrishnan, G. Sankaranarayanan and C. Suyambulingom

Vol. 36 (1971), No. 3, 603–613
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

                      ∑∞    k ∑∞
exp{λt[ϕ(x) − 1]},ϕ(x) =   pkx ,   pk = 1.
k=1     k=1

Lay off to the right of the origin successive intervals of length zj∕jα,j = 1,2, . Explicitly the end points are

t1(z) = 0
tj(z) = k=1j1zk∕kα,j = 2,3,,α > 0,
and
t(z) = k=1zk∕kα
Following Shepp and Lloyd Lr, the length of the r-th longest cycle and Sr, the length of the r-th shortest cycle have been defined for our choice of x(t) and tj,j = 1,2, . This paper obtains the asymptotics for the m-th 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.

Mathematical Subject Classification 2000
Primary: 60C05
Milestones
Received: 23 October 1969
Revised: 14 May 1970
Published: 1 March 1971
Authors
V. Balakrishnan
G. Sankaranarayanan
C. Suyambulingom