Further refinements of Euler-Mahonian statistics for multipermutations

TL;DR

This work extends Euler-Mahonian theory to multipermutations by introducing the $g$-gap $\ell$-level Denert's statistic $g\mathsf{den}_{\ell}$ and the $g$-gap $\ell$-level excedance $g\mathsf{exc}_{\ell}$, as natural generalizations of classical Denert and excedance. It develops two explicit bijections, $\Phi^{\mathsf{den}}_{g,h}$ and $\Phi^{\mathsf{maj}}_{g,\ell}$, built via $\phi_{n,g,h}$ and a $g$-gap labeling, to prove equidistribution: for all $1\le h\le g+\ell$, the pairs $(g\exc_{\ell}, g\den_{h})$ and $(g\des_{\ell}, g\maj_{\ell})$ are equidistributed over multipermutations, generalizing Han's and Liu's results. The authors provide an alternative, bijective verification of the classical equidistribution between $(\mathsf{des},\mathsf{maj})$ and $(\mathsf{exc},\mathsf{den})$, and confirm a Huang-Lin-Yan conjecture. They further show that for multipermutations of $M=\{1^k,\ldots,n^k\}$, the pair $(g\exc_{\ell}, g\den_{h})$ is $r$-Euler-Mahonian with $r=g+\ell-1$, extending Liu's permutation results to the multipermutation setting, thereby broadening the reach of Euler-Mahonian phenomena.

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.

Figures (4)

• Figure 1: An example of Case 2 of the map $\phi_{n,g,h}$.
• Figure 2: An example of Case 3 of the map $\phi_{n,g,h}$.
• Figure 3: An example of Case 1 of the map $\psi_{n,g,h}$.
• Figure 4: An example of Case 2 of the map $\psi_{n,g,h}$.

Theorems & Definitions (34)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Remark 1.4
• Theorem 1.5: Liu Liu3
• Conjecture 1.6: Huang--Lin--Yan Huang-lin-yan2
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 2.1