The Number of Inversions and the Major Index of Permutations are Asymptotically Joint-Independently Normal

TL;DR

This paper proves that the number of inversions and the major index in permutations are asymptotically jointly normal and independent, providing precise formulas for their mixed moments using recurrence relations.

Contribution

It establishes the joint asymptotic normality and independence of two key permutation statistics with detailed moment formulas, advancing understanding in permutation combinatorics.

Findings

Number of inversions and major index are asymptotically joint-normal.

The two statistics are asymptotically independent.

Derived precise asymptotic formulas for mixed moments.

Abstract

We use recurrences (alias difference equations) to prove the longstanding conjecture that the two most important permutation statistics, namely the number of inversions and the major index, are asymptotically joint-independently-normal. We even derive more-precise-than needed asymptotic formulas for the (normalized) mixed moments. This is the fully revised second edition, incorportating the many insightful comments of nine conscientious NON-anonymous referees listed under the authors' names. This article is exclusively published in the on-line journal "Personal Journal of Shalosh B. Ekhad and Doron Zeilberger" and this arxiv.