Further results on $r$-Euler-Mahonian statistics
Kaimei Huang, Sherry H. F. Yan
TL;DR
The paper advances the study of $r$-Euler-Mahonian statistics by proving two main results. First, it confirms Liu's conjecture that $(g\mathsf{exc}_\ell, g\mathsf{den}_\ell)$ is $(g+\ell-1)$-Euler-Mahonian, via a bijection that equidistributes these statistics with $(r\mathsf{des}, r\mathsf{maj})$ where $r=g+\ell-1$, and it recovers the classical $(\mathsf{des}, \mathsf{maj})$ vs. $(\mathsf{exc}, \mathsf{den})$ case when $g=\ell=1$. Second, it extends this paradigm to the pair $(g\mathsf{exc}_\ell, g\mathsf{den}_{g+\ell})$, establishing the analogous $(g+\ell-1)$-Euler-Mahonian property. The authors do this by developing insertion- and labeling-based bijective constructions that generalize Rawlings's and Liu's insertion methods, including a novel extension combining four insertion maps (and variants) to achieve the desired equidistributions. The results imply the Mahonian nature of the Denert-type statistics $g\mathsf{den}_\ell$ and provide a unified bijective framework for several families of $r$-Euler-Mahonian statistics, with potential implications for further exploration of permutation statistics and their distributional symmetries.
Abstract
As natural generalizations of the descent number ($\des$) and the major index ($\maj$), Rawlings introduced the notions of the $r$-descent number ($r\des$) and the $r$-major index ($r\maj$) for a given positive integer $r$. A pair $(\st_1, \st_2)$ of permutation statistics is said to be $r$-Euler-Mahonian if $ (\mathrm{st_1}, \mathrm{st_2})$ and $ (r\des, r\maj)$ are equidistributed over the set $\mathfrak{S}_{n}$ of all permutations of $\{1,2,\ldots, n\}$. The main objective of this paper is to confirm a recent conjecture posed by Liu which asserts that $(g\exc_\ell, g\den_\ell)$ is $(g+\ell-1)$-Euler-Mahonian for all positive integers $g$ and $\ell$, where $g\exc_\ell$ denotes the $g$-gap $\ell$-level excedance number and $g\den_\ell$ denotes the $g$-gap $\ell$-level Denert's statistic. This is accomplished via a bijective proof of the equidistribution of $(g\exc_\ell, g\den_\ell)$ and $ (r\des, r\maj)$ where $r=g+\ell-1$. Setting $g=\ell=1$, our result recovers the equidistribution of $(\des, \maj)$ and $(\exc, \den)$, which was first conjectured by Denert and proved by Foata and Zeilberger. Our second main result is concerned with the analogous result for $(g\exc_\ell, g\den_{g+\ell})$ which states that $(g\exc_\ell, g\den_{g+\ell})$ is $(g+\ell-1)$-Euler-Mahonian for all positive integers $g$ and $\ell$.