  The tetrahedron equation introduced by Zamolodchikov is a three-dimensional
generalization of the Yang-Baxter equation. Several types of solutions to the
tetrahedron equation that have connections to quantum groups can be viewed as
$q$-oscillator valued vertex models with matrix elements of the $L$-operators
given by generators of the $q$-oscillator algebra acting on the Fock space.
Using one of the $q=0$-oscillator valued vertex models introduced by
Bazhanov-Sergeev, we introduce a family of partition functions that admits an
explicit algebraic presentation using Schur functions. Our construction is
based on the three-dimensional realization of the Zamolodchikov-Faddeev algebra
provided by Kuniba-Okado-Maruyama. Furthermore, we investigate an inhomogeneous
generalization of the three-dimensional lattice model. We show that the
inhomogeneous analog of (a certain subclass of) partition functions can be
expressed as loop elementary symmetric functions.
