A sign-reversing involution for an extension of Torelli's Pfaffian identity

TL;DR

This paper extends Torelli's Pfaffian identity by evaluating the hyperpfaffian of certain skew-symmetric polynomials using a sign-reversing involution, revealing a product involving Vandermonde and polynomial coefficients.

Contribution

It introduces a novel involution-based method to evaluate hyperpfaffians of polynomial tensors, generalizing Torelli's classical identity.

Findings

Explicit evaluation of the hyperpfaffian in terms of Vandermonde and polynomial coefficients

Extension of Torelli's identity to higher-order skew-symmetric polynomials

Application of sign-reversing involution technique to combinatorial structures

Abstract

We evaluate the hyperpfaffian of a skew-symmetric $k$-ary polynomial $f$ of degree $k/2 \cdot (n-1)$. The result is a product of the Vandermonde product and a certain expression involving the coefficients of the polynomial $f$. The proof utilizes a sign reversing involution on a set of weighted, oriented partitions. When restricting to the classical case when $k=2$ and the polynomial is $(x_{j} - x_{i})^{n-1}$, we obtain an identity due to Torelli.