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 / 2 ⋅ ( 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.