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

Richard Ehrenborg; N. Bradley Fox

doi:10.1016/j.disc.2014.06.002

paper

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

arXiv:1408.6405 · doi:10.1016/j.disc.2014.06.002

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.

8 pages, 1 figure