Elliptic free-fermion model with OS boundary and elliptic Pfaffians

We introduce and study a class of partition functions of an elliptic free-fermionic face model. We study the partition functions with a triangular boundary using the off-diagonal $K$-matrix at the boundary (OS boundary), which was introduced by Kuperberg as a class of variants of the domain wall boundary partition functions. We find explicit forms of the partition functions with OS boundary using elliptic Pfaffians. We find two expressions based on two versions of Korepin's method, and we obtain an identity between two elliptic Pfaffians as a corollary.