On the Number of Cycles of Given Length of a Free Word in Several Random Permutations

Let w ≠ 1 be a free word in the symbols g1,…, gk and their inverses (i.e., an element of the free group Fk). For any s1,…, sk, in the group sn of all permutation of n objects, we denote by w(s1,…,sk) ϵ Sn the permutation obtained by replacing g1,…, gk with s1,…, sk in the expression of w. Let X (s1,…, sk) denote the number of cycles of length L of w(s1,…, sk). For fixed w and L, we show that X, viewed as a random variable on Snk, has (for n ∞) a Poisson-type limit distribution, which can be computed precisely. © 1994 John Wiley & Sons, Inc.