On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words
Abstract
We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlevé V equations. We show further that in the weakly increasing case the generating function gives the distribution of the smallest eigenvalue in the k x k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N > infinity limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k x k hermitian matrices of trace zero.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 1999
 arXiv:
 arXiv:math/9904042
 Bibcode:
 1999math......4042T
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 05A15;
 47B35;
 60C05;
 82B23
 EPrint:
 30 pages, revised version corrects an error in the statement of Theorem 4