  We prove an explicit expression for the solutions of the discrete Schwarzian
octahedron recurrence, also known as the discrete Schwarzian KP equation
(dSKP), as the ratio of two partition functions. Each one counts weighted
oriented dimer configurations of an associated bipartite graph, and is equal to
the determinant of a Kasteleyn matrix. This is in the spirit of Speyer's result
on the dKP equation, or octahedron recurrence [Spe07]. One consequence is that
dSKP has zero algebraic entropy, meaning that the growth of the degrees of the
polynomials involved is only polynomial. There are cancellations in the
partition function, and we prove an alternative, cancellation free explicit
expression involving complementary trees and forests. Using all of the above,
we show several instances of the Devron property for dSKP, i.e., that certain
singularities in initial data repeat after a finite number of steps. This has
many applications for discrete geometric systems and is the subject of the
companion paper [AdTM22]. We also prove limit shape results analogous to the
arctic circle of the Aztec diamond. Finally, we discuss the combinatorics of
all the other octahedral equations in the classification of Adler, Bobenko and
Suris [ABS12].
