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

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