Alternating Power Difference (APD)

• APD is defined as an integer-valued symmetric group statistic computed from the m-th power sum of trace functions over all permutations of an integer matrix.
• It exhibits a vanishing phenomenon for lower powers until the first appearance degree, where closed-form formulas and combinatorial invariants emerge.
• APD bridges matrix theory, combinatorics, and representation theory by revealing structural symmetries and spectral properties in classical matrix families.

The Alternating Power Difference (APD) is an integer-valued symmetric group statistic associated to a function derived from the trace of a given integer matrix $A$ paired with a permutation. For a matrix $A\in \mathbb{Z}^{n\times n}$ and $\sigma\in S_n$, the APD considers $$f_A(\sigma) := \operatorname{tr}(A P_\sigma) = \sum_{i=1}^n A_{i,\sigma(i)}$$ and sums the $m$-th power of $f_A(\sigma)$ over $S_n$ weighted by the sign, yielding $\APD_m(A) := \sum_{\sigma\in S_n}\sgn(\sigma)f_A(\sigma)^m$. Famously, for all classical matrix families investigated, $\APD_m(A)$ vanishes for a long initial range of $m$, then becomes nonzero at a specific degree $m_1(A)$, the first appearance degree. This phenomenon gives rise to new closed-form combinatorial invariants and connects the APD to foundational questions in matrix theory, combinatorics, and group representation.

1. Definitions and Core Concepts

For $A\in\mathbb{Z}^{n\times n}$ and permutation $\sigma\in S_n$, define

$$f_A(\sigma) := \operatorname{tr}(A P_\sigma) = \sum_{i=1}^n A_{i,\sigma(i)}$$

where $P_\sigma$ is the permutation matrix for $\sigma$. The APD of degree $m$ is

$\APD_m(A) := \sum_{\sigma\in S_n} \sgn(\sigma) f_A(\sigma)^m.$

The first appearance degree is the minimal $m$ for which $\APD_m(A) \neq 0$: $m_1(A) := \min \{ m\ge 1 : \APD_m(A)\neq 0 \}.$ The value $\APD_{m_1}(A)$ is the first nonzero APD and encodes matrix-type dependent algebraic information ((Takemura, 20 Dec 2025), Defs. 3.1–3.2, Eqns. (2)–(4)).

2. Closed-Form Results and Main Conjectures

The values of $m_1(A)$ and $\APD_{m_1}(A)$ have closed forms for several classical matrix families. These are, unless stated as theorems, numerical conjectures, verified computationally for moderate $n$.

Matrix Family	$m_1(A)$	$\APD_{m_1}(A)$
Identity $I_n$	$n-1$	$n!$
Circulant $C_n$	$n-1$	$(-1)^{T_{n-1}} n^{n-2} n!$
Lattice $A^{(2,d)}$	$T_{n-1}$	$(2d)^{T_{n-1}} T_{n-1}! \prod_{k=1}^{n-1} k!$
Hilbert $H_n$	$n-1$	$\det(H_n) n n!$
Multiplication Table $M_n$	$T_{n-1}$	$T_{n-1}! \prod_{k=1}^{n-1} k!$
Vandermonde $V_n$	$n-1$	$(n-1)! \prod_{k=1}^{n-1} k!$
Pascal $P_n$	$n-1$	$(n-1)!$

Here, $T_{n-1} := \frac{1}{2} n (n-1)$, and the Hilbert determinant is given by

$$\det(H_n) = \frac{ \prod_{k=1}^{n-1} (k!)^4 }{ \prod_{k=1}^{2n-1} k! }$$

((Takemura, 20 Dec 2025), §4–§14, Conjs. 4.1–14.2, Thm. 8.1).

A pervasive pattern is $m_1(A) = n-1$ for highly symmetric matrices, shifting to $T_{n-1}$ for certain lattice and multiplication table matrices. For the Vandermonde, $\APD_{n-1}(V_n) = (n-1)! \det(V_n)$, and for Hilbert, $\APD_{n-1}(H_n) = \det(H_n) n n!$ (see Fact 12.1 and Conjs. 8.2–8.3).

3. Algebraic Structure and Derivation Techniques

The APD is computed as a symmetric-group sign-weighted sum over powers of $f_A(\sigma)$. Expansion yields: $\APD_m(f_A) = \sum_\sigma \sgn(\sigma) \left( \sum_i A_{i,\sigma(i)} \right)^m = \sum_{i_1,\dots,i_m} \sum_\sigma \sgn(\sigma) A_{i_1,\sigma(i_1)} \cdots A_{i_m,\sigma(i_m)}.$ This facilitates grouping by conjugacy class type $\lambda\vDash n$, invoking the character-sum identity: $\sum_{\sigma\in S_n}\sgn(\sigma) \prod_j x_{\sigma(j)} = \sum_\lambda \chi^{\text{sgn}}_\lambda z_\lambda^{-1} p_\lambda(x_1,\dots,x_n),$ where $\chi^{\text{sgn}}_\lambda$ is the sign character and $p_\lambda$ the power-sum symmetric functions. For degrees $m<m_1(A)$, the representation-theoretic structure (e.g., eigenvalue multiplicities, lattice symmetries) forces complete cancellation. At $m=m_1(A)$, the first irreducible character survives, producing a closed-form with determinant and factorial factors (see (Takemura, 20 Dec 2025), §3, derivations; Eqns. (6), (14)).

For example, for $A = V_n$ (Vandermonde), $m_1 = n-1$, and the value involves the product $\prod_{k=1}^{n-1} k!$. For $A^{(2,d)}$ and $M_n$, the "superfactorial" product arises.

4. Examples and Explicit Computations

Specialized computations for low $n$ in major classes illustrate the closed-form results:

• Lattice $A^{(2,d)}$, $n=3$: $T_2=3$, $\APD_3 = 96 d^3$.
• Lattice $A^{(2,d)}$, $n=4$: $T_3=6$, $\APD_6 = 552960 d^6$.
• Vandermonde $V_3$: $m_1=2$, $\APD_2=4$; $V_4$: $m_1=3$, $\APD_3=72$.
• Circulant $C_5$: $m_1=4$, $T_4=10$, $5^{3}=125$, so $\APD_4 = 15000$.

These computations illustrate how the APD encapsulates both the structure of the underlying matrix and the combinatorial complexity of the symmetric group ((Takemura, 20 Dec 2025), §4).

5. The First-Appearance Phenomenon

The "first-appearance phenomenon" denotes the APD vanishing identically for all $m < m_1(A)$ and becoming nonzero precisely at $m=m_1(A)$. For $m < m_1(A)$, the multisets of trace values over even and odd permutations are "power-sum indistinguishable" up to degree $m_1-1$; i.e., the phenomenon is a multiset analog of the Prouhet-Tarry-Escott problem.

Equivalently, all irreducible $S_n$-characters cancel up to $m_1-1$, with a unique representation-theoretic contribution emerging at $m_1$. This reflects a minimal moment where the discrepancy between the even and odd classes in the trace function becomes detectable ((Takemura, 20 Dec 2025), §5).

6. Connections, Applications, and Further Significance

The APD links additive and multiplicative invariants of matrices, unifying classical determinants with group-theoretic character sums. For $V_n$ and $H_n$, $\APD_{n-1}(A)$ is essentially proportional to the determinant, up to factorial factors (Conjs. 8.2, 12.3).

The APD serves as a new, spectral-sensitive, group-theoretic invariant of $A$, responsive to features like circulant shifts, lattice parameters, and combinatorial symmetries in $A$'s construction. Potential applications include:

• Detection of hidden combinatorial symmetries in design matrices.
• Construction of Prouhet-Tarry-Escott (PTE) partitions using trace statistics.
• Investigation of connections between symmetric-group characters and matrix invariants.

The emergence of superfactorials (e.g., $\prod_{k=1}^{n-1}k!$) in many formulas suggests connections with classical group volumes, the hook-length formula, and certain topological invariants. This further indicates possible links between the APD and concepts in algebraic combinatorics and representation theory ((Takemura, 20 Dec 2025), §6).