Let $n$ be a positive even integer. We introduce three types of skew-symmetric matrices, $A_{n+2}$, $B_n$ and $C_n$ (the subscript denotes the dimension of the matrices) in terms of the variables $z_1,z_2,\ldots,z_{n+2}$. The off-diagonal elements are given by $$ (A_{n+2})_{i,j} = \frac{1}{z_i-z_j}\ , \\ (B_{n})_{i,j} = \frac{1}{2}\frac{(z_i-z_{n+1})^2(z_j-z_{n+2})^2+(z_i-z_{n+2})^2(z_j-z_{n+1})^2)}{z_i-z_j}\ , \\ (C_{n})_{i,j} = \frac{(z_i-z_{n+1})(z_i-z_{n+2})(z_j-z_{n+1})(z_j-z_{n+2})}{z_i-z_j} \ . $$
One then has the following identity of symmetric polynomials $$ \rm{Pf} (A_{n+2}) \prod_{1\leq i < j \leq n+2}(z_i-z_j)= \Bigl( 2\,\rm{Pf} (B_{n}) - \rm{Pf} (C_{n}) \Bigr) \prod_{1\leq i < j \leq n}(z_i-z_j) \ . $$ This identity was obtained in a rather roundabout way using (the Ising) conformal field theory. One can also prove this identity by induction, which is not very enlightening.
I would like to know if there is a more interesting combinatorial derivation or interpretation of this identity.
