Alternating Power Difference and Matrix Symmetry: Closed-Form Formulas for the First Appearance Degree $m_1$

This paper focuses on an integer-valued function $f_A(\sigma) := \operatorname{tr}(A P_\sigma)$ defined uniformly from a specific square matrix $A$ of order $n$ and a permutation $\sigma$ on the symmetric group $S_n$. The main objective of this study is to investigate in detail the algebraic behavior of the Alternating Power Difference (APD), denoted as $APD_m(f_A)$, and its first appearance degree $m_1(f_A)$ for this function $f_A$ 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 $APD_m(A)$ 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 $m_1(A)$ and the first appearance value $APD_{m_1}(A)$, 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.