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

We evaluate the hyperpfaffian of a skew-symmetric $k$-ary polynomial $f$ of degree $k\mathrm{/}2\cdot (n-1)$. 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.