1. Introduction

Consider the length of a longest increasing subsequence in a permutation is an increasing subsequence of length r. If we give SN the uniform probability distribution, becomes a random variable and we want to investigate its distribution. This problem was first addressed by Ulam [1961], who made Monte Carlo simulations and concluded that the expectation E[lN] seems to be of order y/N. The first rigorous result was obtained by Hammersley [1972], who considered the following variant of the problem. Consider a Poisson process in the square [0,1] x [0,1] with intensity α, so that the number M of points in the square is Poisson distributed with mean α. Let and be the x- and y-coordinates of the points in the square. This associates a permutation α ∈ SM with each point configuration, and if we condition M to be fixed, equal to iV say, we get the uniform distribution on SN We see that IM(α) equals the number of points, L(α), in an up/right path from (0,0) to (1,1) through the points, and containing as many points as possible.