The authors consider the length, lN , of the length of the longest increasing subsequence of a random permutation of N numbers. The main result in this paper is a proof that the distribution function for lN , suitably centered and scaled, converges to the Tracy-Widom distribution [TW1] of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is ...

We show that a wide variety of generalized increasing subsequence problems admit a one parameter family of extensions for which we can exactly compute the mean length of the longest increasing subsequence. By the nature of the extension, this gives upper bounds on the mean in the unextended model, which turn out to be asymptotically tight for all of the models that have so far been analyzed. A ...

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i.e., labellings of the nodes by distinct integers of the set {1, . . . , n} in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: recurs...

This paper is concerned with certain connections between the ensemble of n × n unitary matrices—specifically the characteristic function of the random variable tr(U)—and combinatorics—specifically Ulam’s problem concerning the distribution of the length of the longest increasing subsequence in permutation groups—and the appearance of Painlevé functions in the answers to apparently unrelated que...

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has been a fruitful source of interesting mathematical phenomena. In this paper we develop a connection between the construction of Busemann functions in the Hammersley last-passage percolation model with i.i.d. random weights, and the existence, ergodicity and uniqueness of equilibrium (or ti...