An interesting new Mahonian permutation statistic

TL;DR

This paper introduces a new Mahonian permutation statistic derived from a standard permutation generation algorithm, showing it is distinct from known statistics and exploring its joint distribution with Eulerian statistics.

Contribution

It presents a novel Mahonian permutation statistic and constructs an Eulerian partner, expanding the understanding of permutation statistics and their distributions.

Findings

The new statistic is Mahonian and distinct from known ones.

Its joint distribution with Eulerian statistics differs from known pairs.

A new Eulerian partner statistic is constructed, forming an Euler-Mahonian pair.

Abstract

The standard algorithm for generating a random permutation gives rise to an obvious permutation statistic $\stat$ that is readily seen to be Mahonian. We give evidence showing that it is not equal to any previously published statistic. Nor does its joint distribution with the standard Eulerian statistics $\des$ and $\exc$ appear to coincide with any known Euler-Mahonian pair. A general construction of Skandera yields an Eulerian partner $\ska$ such that $(\ska, \stat)$ is equidistributed with $(\des, \maj)$. However $\ska$ itself appears not to be a known Eulerian statistic. Several ideas for further research on this topic are listed.