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The canonical sylvan resolution is a resolution of an arbitrary monomialideal over a polynomial ring that is minimal and has an explicit combinatorialformula for the differential. The differential is a weighted sum over latticepaths of weights of chain-link fences, which are sequences of faces that arelinked to each other via higher-dimensional analogues of spanning trees. Alonga lattice path in the three-variable case, these weights can be condensed to asingle weight contributing to the combinatorial formula for the differentialthat bypasses any computation of chain-link fences. The main results in thispaper express the sylvan matrix entries for monomial ideals in three variablesas a sum over lattice paths of simpler weights that depend only on the numberof specific Koszul simplicial complexes that lie along the correspondinglattice path. Certain entries have numerators equal to the number of latticepaths in $\mathbb{N}^2$ that follow specific restrictions.

Enumerative and planar combinatorics of trivariate monomial resolutions