Random Unitary Matrices, Permutations and Painlevé

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 questions. Among the results is a representation in terms of a Painlevé V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painlevé II function for the limiting distributiuon of the length of the longest increasing subsequence in the hyperoctahedral groups.