  In testing the independence of two Gaussian populations, one computes the
distribution of the sample canonical correlation coefficients, given that the
actual correlation is zero. The "Laplace transform" of this distribution is not
only an integral over the Grassmannian of p-dimensional planes in complex
n-space, but is also related to a generalized hypergeometric function. Such
integrals are solutions of Painlev\'e-like equations. They also have
expansions, related to random words of length l formed with an alphabet of p
letters. Given that each letter appears in the word, the maximal length of the
disjoint union of p increasing subsequences of the word clearly equals l. But
the maximal length of the disjoint union of p-1 increasing subsequences leads
to a non-trivial distribution. It is precisely this probability which appears
in the expansion above.
