Further refinements of Euler-Mahonian statistics for multipermutations

Abstract

Permutation statistics constitute a classical subject of enumerative combinatorics. In her study of the genus zeta function, Denert discovered a new Mahonian statistic for permutations, which is called the Denert's statistic ({\bf $\den$}) by Foata and Zeilberger. As natural extensions of the $r$-descent number ({\bf $r\des$}) and the $r$-major index ({\bf $r\maj$}) introduced by Rawlings, Liu introduced the $g$-gap $\ell$-level descent number ({\bf $g\des_{\ell}$}) and the $g$-gap $\ell$-level major index ({\bf $g\maj_{\ell}$}) for permutations. In this paper, we introduce the $g$-gap $\ell$-level Denert's statistic ({\bf $g\den_{\ell}$}) and the $g$-gap $\ell$-level excedance number ({\bf $g\exc_{\ell}$}) for multipermutations, which serve as natural generalizations of the Denert's statistic ({\bf $\den$}) and the excedance number ({\bf $\exc$}) for multipermutations first introduced by Han. By constructing two explicit bijections, we establish the equidistribution of the pairs $(g\exc_{\ell}, g\den_{h} )$ and $(g\des_{\ell}, g\maj_{\ell})$ over multipermutations for all $1\leq h\leq g+\ell$. Our result provides a new proof of the equidistribution of the pairs ($\des$, $\maj$) and ($\exc$, $\den$) over multipermutations originally derived by Han and enables us to confirm a recent conjecture posed by Huang-Lin-Yan. Furthermore, we demonstrate that for all $1\leq h\leq g+\ell$, the pair $(g\exc_\ell, g\den_{h})$ is $r$-Euler-Mahonian over multipermutations of $M=\{1^k, 2^k, \ldots, n^k\}$ where $r=g+\ell-1$ and $k\geq 1$, which extends a recent novel result derived by Liu from permutations to multipermutations.