New $r$-Euler--Mahonian statistics involving Denert's statistic

TL;DR

This paper introduces new $r$-Euler--Mahonian permutation statistics, provides bijective proofs of classical and extended results, and unifies various $r$-Euler--Mahonian classes through a generalized bijection.

Contribution

It offers a new bijective proof for the classical Euler--Mahonian property and extends it to $r$-Euler--Mahonian statistics, unifying multiple classes under a generalized framework.

Findings

Proved $( extsf{exc}, extsf{den})$ is Euler--Mahonian via a new bijection.

Extended the bijection to show $( extsf{exc}_{r}, extsf{den})$ is $r$-Euler--Mahonian.

Established a unified framework for all known $r$-Euler--Mahonian statistics.

Abstract

Recently, we proved the equidistribution of the pairs of permutation statistics $(r\textsf{des},r\textsf{maj})$ and $(r\textsf{exc},r\textsf{den})$. Any pair of permutation statistics that is equidistributed with these pairs is said to be $r$-Euler--Mahonian. Several classes of $r$-Euler--Mahonian statistics were established by Huang--Lin--Yan and Huang--Yan. Inspired by their bijections, we provide a new bijective proof of the classical result that $(\textsf{exc},\textsf{den})$ is Euler--Mahonian. Using this bijection, we further show that $(\textsf{exc}_{r},\textsf{den})$ is $r$-Euler--Mahonian, where $\textsf{exc}_{r}$ denotes the number of $r$-level excedances (i.e., excedances at least $r$). Furthermore, by extending our bijection, we establish a more general result that encompasses all the aforementioned results.