On a conjecture concerning the r-Euler-Mahonian statistic on permutations

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A Pub Date : 2025-01-17 DOI:10.1016/j.jcta.2025.106008

Kaimei Huang , Zhicong Lin , Sherry H.F. Yan

引用次数: 0

Abstract

A pair

(s t_{1}, s t_{2})

of permutation statistics is said to be r-Euler-Mahonian if

(s t_{1}, s t_{2})

and (rdes, rmaj) are equidistributed over the set

S_{n}

of all permutations of

{1, 2, \dots, n}

, where rdes denotes the r-descent number and rmaj denotes the r-major index introduced by Rawlings. The main objective of this paper is to prove that

({exc}_{r}, {den}_{r})

and (rdes, rmaj) are equidistributed over

S_{n}

, thereby confirming a recent conjecture posed by Liu. When

r = 1

, the result recovers the equidistribution of

(des, maj)

and

(exc, den)

, which was first conjectured by Denert and proved by Foata and Zeilberger.

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关于r-Euler-Mahonian排列统计量的一个猜想

如果（st1,st2）和（rdes, rmaj）在{1,2，…，n}的所有排列的集合Sn上是等分布的，我们称一对（st1,st2）排列统计量为r-Euler-Mahonian，其中rdes表示r-下降数，rmaj表示罗林斯引入的r-主指数。本文的主要目的是证明（excr,denr）和（rdes, rmaj）在Sn上是均匀分布的，从而证实Liu最近提出的一个猜想。当r=1时，结果恢复了（des,maj）和（exc,den）的均一分布，该均一分布最早由Denert推测，并由Foata和Zeilberger证明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Combinatorial Theory Series A 数学-数学

CiteScore

2.90

自引率

9.10%

发文量

审稿时长

12 months

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.