Permutations with roots

We prove that the probability p2(n) that a random permutation of length n has a square root is monotonically nonincreasing in n. More generally, we prove that the probability pr(n) that a random permutation of length n has an rth root, r prime, is monotonically nonincreasing in n. We also show for all r≥2 that pr(n)0 as n∞. While doing this, we combinatorially prove that pr(n)=pr(n+1) for r prime and for all n not congruent to −1 mod r, and we construct several bijections for sets of permutations defined by modular class restrictions on the cycle lengths. We also include a simple probabilistic proof that, for r≥2, pr(n)0 as n∞. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 157–167, 2000