Random unitary matrices, permutations and Painleve

This paper is concerned with certain connections between the ensemble of n x 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 Painleve functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Balk, Deift and Johansson) an expression in terms of a Painleve' II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups.