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

Abstract

Recently, we proved the equidistribution of the pairs of permutation statistics $(r\text{des},r\text{maj})$ and $(r\text{exc},r\text{den})$. 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 $(\text{exc},\text{den})$ is Euler--Mahonian. Using this bijection, we further show that $({\text{exc}}_r,\text{den})$ is $r$-Euler--Mahonian, where ${\text{exc}}_r$ 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.