  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.
