Enumerative and planar combinatorics of trivariate monomial resolutions

TL;DR

This paper provides a combinatorial framework for explicitly computing the differentials in the sylvan resolution of monomial ideals in three variables, simplifying calculations by reducing complex lattice path weights.

Contribution

It introduces a simplified combinatorial formula for the sylvan resolution's differentials in three-variable monomial ideals, avoiding complex chain-link fence computations.

Findings

Matrix entries expressed as sums over lattice paths with simplified weights

Numerators correspond to counts of restricted lattice paths in \\mathbb{N}^2

Explicit combinatorial formulas facilitate calculations of resolutions

Abstract

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