The Mahonian probability distribution on words is asymptotically normal

TL;DR

This paper proves that the Mahonian distribution on words approaches a normal distribution asymptotically, using both computational and analytical methods, and explores related local limit theorems and properties like log-concavity.

Contribution

It provides two proofs of asymptotic normality for the Mahonian distribution, including a computer-assisted method and an analytical approach, and investigates local limit theorems and coefficient properties.

Findings

Distribution is asymptotically normal, confirmed by two different proofs.

The coefficients of the q-multinomial are log-concave near the center.

A conjecture is proposed for further work on local limit theorems.

Abstract

The Mahonian statistic is the number of inversions in a permutation of a multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting function for this statistic is the $q$ analog of the multinomial coefficient $\binom{a_1+...+a_m}{a_1,... a_m}$, and the probability generating function is the normalization of the latter. We give two proofs that the distribution is asymptotically normal. The first is {\it computer-assisted}, based on the method of moments. The Maple package {\tt MahonianStat}, available from the webpage of this article, can be used by the reader to perform experiments and calculations. Our second proof uses characteristic functions. We then take up the study of a local limit theorem to accompany our central limit theorem. Here our result is less general, and we must be content with a conjecture about further work. Our local limit theorem permits us to conclude that the coeffiecients of the $q$-multinomial are log-concave, provided one stays near the center (where the largest coefficients reside.)