Further refinements of Euler-Mahonian statistics for multipermutations

TL;DR

This paper introduces new Euler-Mahonian statistics for multipermutations, proves their equidistribution via explicit bijections, and extends known permutation results to the multipermutation setting, confirming recent conjectures.

Contribution

It defines generalized Denert's and excedance statistics for multipermutations and establishes their equidistribution with descent and major index statistics, extending classical permutation results.

Findings

Proved equidistribution of new statistic pairs over multipermutations.

Provided a new proof of classical permutation statistic equidistribution.

Extended Euler-Mahonian results from permutations to 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.