  Given uniform probability on words of length M=Np+k, from an alphabet of size
p, consider 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 \leq M-N . A generating function for this
probability has the form of an integral over the Grassmannian of p-planes in
complex C^n. The present paper shows that the asymptotics of this probability,
when N tends to infinity, is related to the kth moment of the
chi^2-distribution of parameter 2p^2. This is related to the behavior of the
integral over the Grassmannian Gr(p,C^n) of p-planes in C^n, when the dimension
of the ambient space C^n becomes very large. A different scaling limit for the
Poissonized probability is related to a new matrix integral, itself a solution
of the Painlev\'e IV equation. This is part of a more general set-up related to
the Painlev\'e V equation.
