bibkey: takemura2025apd authors: Kenichi Takemura year: 2025 title: “Alternating Power Difference and Matrix Symmetry: Closed-Form Formulas for the First Appearance Degree m_1” doi: 10.48550/arXiv.2512.18169 url: https://arxiv.org/abs/2512.18169v1 claim: “Definition 1 gives APD_m(f) = sum over sigma in S_n of sgn(sigma) f(sigma)^m. Definition 2 calls the least m >= 1 with nonzero APD the first appearance degree. The fixed-point function is the number of fixed points of sigma. Conjecture 1 states m_1(I_n) = n - 1 for n >= 2, and Conjecture 2 states APD_{n-1}(I_n) = n!.” strata_touched:

• D5/S3/ArithSums/TakemuraFixedPointAlternatingPowerDifference license: citation-only triage: anchor

Alternating Power Difference and Matrix Symmetry

Kenichi Takemura’s arXiv version 1 defines the alternating power difference over the symmetric group and the first appearance degree. Section 5.2.1 identifies the identity-matrix function with the number of fixed points of a permutation. The paper records the two closed forms as Conjectures 1 and 2: the first appearance degree is n - 1, and the first appearance value is n!, for n >= 2.

The formal development expands powers of the fixed-point count into tuple sums, evaluates the signed sum over permutations fixing an image set, and counts the surviving injections.

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• DOI: 10.48550/arXiv.2512.18169
• URL: https://arxiv.org/abs/2512.18169v1