Probability generating functions for Sattolos algorithm

Abstract. 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. H. 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 cor-rectness, we generalize the abovementioned probabilistic results by determining the “grand ” probability generating functions of the ran-dom 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 pi ∈ Sn on i ∈ [n] is denoted by i · pi