Enumerative proof of a curious congruence for Eulerian numbers

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics Pub Date : 2025-10-09 DOI:10.1016/j.aam.2025.102977

Xiangzi Meng , Hao Pan

引用次数: 0

Abstract

The Eulerian number

〈 \begin{matrix} n \\ k \end{matrix} 〉

counts all permutations on

{0, 1, \dots, n - 1}

having exactly k ascents. In this paper, we give an enumerative proof of the following congruence:

〈 \begin{matrix} a p - 1 \\ b p + l \end{matrix} 〉 \equiv {(- 1)}^{b} {(l + 1)}^{a - 1} (\begin{matrix} a - 1 \\ b \end{matrix}) (\mod p),

where p is prime,

0 \leq b < a

and

0 \leq l \leq p - 1

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欧拉数的一个奇异同余的枚举证明

欧拉数< nk >计算{0,1，…，n−1}上的所有恰好有k个上升的排列。本文给出了下列同余的一个枚举证明：< ap−1bp+l >≡(−1)b(l+1)a−1(a−1b)(modp)，其中p为素数，0≤b<；a且0≤l≤p−1。

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

Advances in Applied Mathematics 数学-应用数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

85 days

期刊介绍： Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas. Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.