A Combinatorial Bijective Proof of the Cauchy Determinant (2023). PDF
The Cauchy Determinant (or the Cauchy Matrix) is a determinantal identity, which implies the famous Cauchy identity for Schur symmetric functions. The Cauchy Determinant can be proved algebraicly by counting poles and zeros and comparing leading coefficients. However, there seems to lack a combinatorial bijective proof. We present a bijective proof of the Cauchy matrix identity, inspired by Gessel's bijective proof of the Vandermonde determinant.