A Simple Proof that Major Index and Inversions are Equidistributed

TL;DR

This paper provides a concise proof demonstrating that the distribution of the major index and the number of inversions in permutations are identical, confirming their equidistribution.

Contribution

It offers a simple, elegant proof of MacMahon's result relating major index and inversions in permutations.

Findings

Major index and inversions are equidistributed in permutations.

The proof simplifies understanding of permutation statistics.

Reinforces classical combinatorial results with a new approach.

Abstract

We present a short proof of MacMahon's classic result that the number of permutations with $k$ inversions equals the number whose major index (sum of positions at which descents occur) is $k$