The bias of three pseudo-random shuffles

Abstract

Three schemes for shuffling a deck ofn cards are studied, each involving a random choice from [n]ⁿ. The shuffles favor some permutations over others sincen! does not dividen ⁿ. The probabilities that the shuffles lead to some simple permutations, for instance cycles left and right and the identity, are calculated. Some inequalities are obtained which lead to information about the least and most likely permutations. Numbers of combinatorial interest occur, notably the Catalan numbers and the Bell numbers.