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

Kenichi Takemura

TL;DR

APD is introduced as an alternating-sum invariant for functions on the symmetric group derived from matrices via f_A(σ) = tr(A P_σ). The work systematically studies the first appearance degree m_1 across diverse matrix families (identity, circulant, Hilbert, Vandermonde, Pascal, and lattice-based matrices), uncovering conjectured closed-form formulas such as m_1(A) = n-1 and explicit APD_{m_1}(A) expressions involving determinants and factorial products. A unifying perspective emerges through constructs like V_Core and superfactorials, and links to the Prouhet-Tarry-Escott problem, suggesting deep symmetry structures governing APD. The study combines extensive numerical verification with rigorous formulations to guide future proofs and broader explorations in matrix theory and combinatorics.

Abstract

This paper focuses on an integer-valued function $f_A(σ) := \operatorname{tr}(A P_σ)$ defined uniformly from a specific square matrix $A$ of order $n$ and a permutation $σ$ 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.

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

TL;DR

APD is introduced as an alternating-sum invariant for functions on the symmetric group derived from matrices via f_A(σ) = tr(A P_σ). The work systematically studies the first appearance degree m_1 across diverse matrix families (identity, circulant, Hilbert, Vandermonde, Pascal, and lattice-based matrices), uncovering conjectured closed-form formulas such as m_1(A) = n-1 and explicit APD_{m_1}(A) expressions involving determinants and factorial products. A unifying perspective emerges through constructs like V_Core and superfactorials, and links to the Prouhet-Tarry-Escott problem, suggesting deep symmetry structures governing APD. The study combines extensive numerical verification with rigorous formulations to guide future proofs and broader explorations in matrix theory and combinatorics.

Abstract

This paper focuses on an integer-valued function defined uniformly from a specific square matrix of order and a permutation on the symmetric group . The main objective of this study is to investigate in detail the algebraic behavior of the Alternating Power Difference (APD), denoted as , and its first appearance degree for this function across various matrix classes. Specifically, we address special matrices such as shifted -th power lattices, Vandermonde matrices, and circulant matrices, analyzing the phenomenon where the value of remains zero as increases until a specific degree (the first appearance phenomenon). In particular, we explore closed-form formulas for the first appearance degree and the first appearance value , 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.

Paper Structure

This paper contains 56 sections, 2 theorems, 58 equations, 10 tables.

Table of Contents

  1. Abstract
  2. Motivation and Background
  3. What is Alternating Power Difference?
  4. Functions Derived from Matrices and APD
  5. Notation Conventions
  6. Identity Matrix of Order n and APD
  7. Definition of n-th Order Identity Matrix
  8. Function $\operatorname{fix}$ Corresponding to Identity Matrix $I_n$
  9. Definition of Fixed Point Function $\operatorname{fix}$
  10. Table of First Appearance Degrees (n=1 to 10)
  11. Conjectured Closed-Form Formulas for First Appearance Degree
  12. Standard Circulant Matrix of Order n and APD
  13. Definition of Standard Circulant Matrix
  14. Table of First Appearance Degrees (n=1 to 10)
  15. Conjectured Closed-Form Formulas for First Appearance Degree
  16. ...and 41 more sections

Key Result

Proposition 1

When $n \geq 2$, the permutation sum $f(\sigma)$ in the lattice with $r=1$ is equal to a constant $C$, and the alternating power difference is zero for all $m \geq 1$. Therefore, the first appearance degree is infinite. That is,

Theorems & Definitions (36)

  • Definition 1: Alternating Power Difference
  • Definition 2: First Appearance Degree
  • Definition 3: Identity Matrix
  • Conjecture 1: First Appearance Degree of Identity Matrix
  • Conjecture 2: Formula for First Appearance Value of Identity Matrix
  • Definition 4: Standard Circulant Matrix
  • Conjecture 3: Coincidence of First Appearance Degrees
  • Conjecture 4: Formula for First Appearance Value of Standard Circulant Matrix
  • Definition 5: n-th Order Row-Shifted r-th Power Lattice
  • Remark 1: Aspect as a Hankel Matrix
  • ...and 26 more