The inversion number and the major index are asymptotically jointly normally distributed on words

TL;DR

This paper proves that the inversion number and the major index are jointly asymptotically normally distributed on words, extending known results from permutations to more general word structures.

Contribution

It establishes the joint asymptotic normality of the inversion number and the major index on words, filling a gap in the understanding of Mahonian statistics.

Findings

Joint distribution converges to a bivariate normal

Extends asymptotic normality results from permutations to words

Provides a comprehensive understanding of Mahonian statistics on words

Abstract

In a recent paper, Baxter and Zeilberger show that the two most important Mahonian statistics, the inversion number and the major index, are asymptotically independently normally distributed on permutations. In another recent paper, Canfield, Janson and Zeilberger prove the result, already known to statisticians, that the Mahonian distribution is asymptotically normal on words. This leaves one question unanswered: What, asymptotically, is the joint distribution of the inversion number and the major index on words? We answer this question by establishing convergence to a bivariate normal distribution.