Probability generating functions for Sattolo’s algorithm

In 1986 S. Sattolo introduced a simple algorithm for uniform random generation of cyclic permutations on a fixed number of symbols. Recently, H. Prodinger analysed two important random variables associated with the algorithm, and found their mean and variance. Mahmoud extended Prodinger’s analysis by finding limit laws for the same two random variables. The present article, starting from the definition of the algorithm, is completely self-contained. After giving a simple new proof of correctness, we generalize the abovementioned probabilistic results by determining the “grand” probability generating functions of the random variables. The focus throughout is on using standard methods that give a unified approach, and open the door to further study. 1 Sattolo’s algorithm For each n ≥ 1, we denote by Sn the symmetric group on the set [n] := {1, . . . , n}. The action of π ∈ Sn on i ∈ [n] is denoted by i · π. Let Cn be the set of n-cycles of Sn (recall that an element of Sn is an n-cycle if and only if its action on [n] has a single orbit). When n = 1, our convention is Cn = Sn. Sattolo [Sat86] introduced the following algorithm for uniform random generation of an element of Cn. Start with the arrangement 1, . . . , n (corresponding to the identity permutation). There are n−1 steps. At the ith step, an element is chosen uniformly at random