Another proof of Wilmes' conjecture

TL;DR

This paper provides a new proof for the monomial case of Wilmes' conjecture, linking Betti numbers of G-parking function ideals to poset topology and maximal parking functions.

Contribution

It introduces a novel proof method using poset topology for the monomial case of Wilmes' conjecture, expanding understanding of Betti numbers in algebraic combinatorics.

Findings

Proof confirms the monomial case of Wilmes' conjecture

Connects Betti numbers to the topology of lcm-lattices

Utilizes a theorem linking Betti numbers to poset topology

Abstract

We present a new proof of the monomial case of Wilmes' conjecture, which gives a formula for the coarsely-graded Betti numbers of the G-parking function ideal in terms of maximal parking functions of contractions of G. Our proof is via poset topology and relies on a theorem of Gasharov, Peeva, and Welker that connects the Betti numbers of a monomial ideal to the topology of its lcm-lattice.