2025/12/20 by Kenichi Takemura, Takemura, Kenichi
Computer Science · Mathematics · #05E05 #11B83 #11D99 #15A15 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Number Theory (math.NT) #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.2512.18169
openalex publication_date 2025/12/20 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28
This paper focuses on an integer-valued function fA(σ) := tr(A Pσ) defined uniformly from a specific square matrix A of order n and a permutation σ on the symmetric group Sn. The main objective of this study is to investigate in detail the algebraic behavior of the Alternating Power Difference (APD), denoted as APDm(fA), and its first appearance degree m1(fA) for this function fA across various matrix classes. Specifically, we address special matrices such as shifted r-th power lattices, Vandermonde matrices, and circulant matrices, analyzing the phenomenon where the value of APDm(A) remains zero as m increases until a specific degree (the first appearance phenomenon). In particular, we explore closed-form formulas for the first appearance degree m1(A) and the first appearance value APDm1(A), presenting Conjectures that hold across multiple matrix classes. These results suggest a deep relationship between the structure of matrices and the analytical properties of functions on the symmetric group, providing new perspectives in matrix theory and combinatorics.