  We compute the limiting distribution, as n approaches infinity, of the number
of cycles of length between gamma n and delta n in a permutation of [n] chosen
uniformly at random, for constants gamma, delta such that 1/(k+1) <= gamma <
delta <= 1/k for some integer k. This distribution is supported on {0, 1, ...
k} and has 0th, 1st, ..., kth moments equal to those of a Poisson distribution
with parameter log (delta/gamma). For more general choices of gamma, delta we
show that such a limiting distribution exists, which can be given explicitly in
terms of certain integrals over intersections of hypercubes with half-spaces;
these integrals are analytically intractable but a recurrence specifying them
can be given. The results herein provide a basis of comparison for similar
statistics on restricted classes of permutations.
