The Number of Cycles of Specified Normalized Length in Permutations

We compute the limiting distribution, as n → ∞, of the number of cycles of length between γn and δn in a permutation of [n] chosen uniformly at random, for constants γ, δ such that 1/(k + 1) ≤ γ < δ ≤ 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 δ γ . For more general choices of γ, δ 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. The distribution of the number of k-cycles in a permutation of [n], for a fixed k, converges to a Poisson distribution with mean 1/k as k → ∞. In particular the mean number of k-cycles and the variance of the number of k-cycles are both 1/k whenever n ≥ k and n ≥ 2k respectively. If instead of holding k constant we let it vary with n, the number of αn-cycles in permutations of [n] approaches zero as n→∞ with α fixed. So to investigate the number of cycles of long lengths, we must rescale and look at many cycle lengths at once. In particular, we consider the number of cycles with length in some interval [γn, δn] as n → ∞. The expectation of the number of cycles with length in this interval is ∑δn k=γn 1/k, which approaches the constant log δ/γ as n grows large. By analogy with the fixed-k case we might expect the number of cycles with length in this interval to be Poisson-distributed. But this cannot be the case, because there is room for at most 1/γ cycles of length at least γn, and the Poisson distribution can take arbitrarily large values. In the case where 1/γ and 1/δ lie in the same interval [1/(k + 1), 1/k] for some integer k, the limit distribution has the same first k moments as Poisson(log δ/γ). For general γ and δ the situation is considerably more complex but a limit distribution still exists. Date: September 15, 2009. 2000 Mathematics Subject Classification. 05A16, 60C05. 1 ar X iv :0 90 9. 29 09 v1 [ m at h. C O ] 1 6 Se p 20 09