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Another proof of Wilmes’ conjecture. (English) Zbl 1283.05215
Summary: 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 V. Gasharov et al. [Math. Res. Lett. 6, No. 5–6, 521–532 (1999; Zbl 0970.13004)] that connects the Betti numbers of a monomial ideal to the topology of its lcm-lattice.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 13D02 Syzygies, resolutions, complexes and commutative rings
##### Keywords:
chip-firing; $$G$$-parking function; lcm-lattice
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##### References:
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