The question of how many times a deck of cards must be shuffled so as to ensure that the resulting distribution of cards corresponds to a draw from the uniform distribution over all permutations of 52 cards has received some attention in the American Press. The present paper continues this line of inquiry. Mathematically, the subject is the study of various kinds of random walks over the permutation group \(S_n\). This paper develops formulae for a variety of functions of permutations resulting from a random walk associated with a repeated riffle shuffle. The main results are obtained by using a bijection theorem of Gessel. The authors provide a self-contained proof of the theorem from first principles.