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

Recently, we proved the equidistribution of the pairs of permutation statistics (rdes,rmaj) and (rexc,rden). 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 (exc,den) is Euler--Mahonian. Using this bijection, we further show that (excᵣ,den) is r-Euler--Mahonian, where excᵣ 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.