Bernoulli Trials and Mahonian Statistics: A Tale of Twoq's

Abstract

Any casual comparison of the topic of Bernoulli trials with that of Mahonian statistics would lead one to the superficial observation that the letter q is common to both subjects: As is well known, the q of the Bernoulli coin-tossing scheme denotes the probability that tails occurs. Somewhat less well known is the fact that Mahonian statistics give rise to a branch of combinatorics that in some circles is referred to as q-combinatorics. Of course, the use of the same symbol in these two contexts is mere coincidence. However, as will be explained, there is a surprising natural connection between these two famous of mathematics. Our discovery of this connection was sparked by a comment made by Professor Jim Delany. Following a colloquium talk on permutation statistics [8], Delany quizzically remarked that there were some striking similarities between the talk and an article by Moritz and Williams [6] that had just appeared in this MAGAZINE. In brief, Moritz and Williams considered the problem of determining the probability that a given permutation would be the result of a certain coin-tossing game. It is their game that forms the link between the Bernoulli coin-tossing scheme and the subject of Mahonian statistics. Beyond being just a curiosity, there are a few practical aspects of our discovery. For starters, the relationship between Bernoulli trials and Mahonian statistics immediately provides the answers to some of the questions posed by Moritz and Williams. We will also make use of the combinatorics of a certain Mahonian statistic, known as the comajor index, to give alternate proofs to some of the results in [6]. We begin our tale of two q's with a short review of Moritz' and Williams' work. Before proceeding, the reader may find it helpful to peruse their article.