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We compute the limiting distribution, as n approaches infinity, of the numberof cycles of length between gamma n and delta n in a permutation of [n] chosenuniformly 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 distributionwith parameter log (delta/gamma). For more general choices of gamma, delta weshow that such a limiting distribution exists, which can be given explicitly interms of certain integrals over intersections of hypercubes with half-spaces;these integrals are analytically intractable but a recurrence specifying themcan be given. The results herein provide a basis of comparison for similarstatistics on restricted classes of permutations.

The number of cycles of specified normalized length in permutations