  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 Baik, 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 group.
