Abstract
Testing the independence of two Gaussian populations involves the distribution of the sample canonical correlation coefficients, given that the actual correlation is zero. The “Laplace transform” (as a function of x) of this distribution is not only an integral over the Grassmannian Gr(p,Fn) of p-dimensional planes in real, complex or quaternion n-space Fn, but is also related to a generalized hypergeometric function. Such integrals are solutions of Painlevé-like equations; in the complex case, they are solutions to genuine Painlevé equations. These integrals over Gr(p,Cn) have remarkable expansions in x, related to random words of length ℓ formed with an alphabet of p letters 1,…, p. The coefficients of these expansions are given by the probability that a word (i) contains a subsequence of letters p,
p−1,…,1 in that order and (ii) that the maximal length of the disjoint union of p−1 increasing subsequences of the word is ⩽k, where k refers to the power of
x. Note that, if each letter appears in the word, then the maximal length of the disjoint union of p increasing subsequences of the word is automatically =ℓ and is thus trivial.