Séminaire Lotharingien de Combinatoire, 84B.85 (2020), 12 pp.

Caroline Klivans and Patrick Liscio

Results in Labeled Chip-Firing

Abstract. In 2016, Hopkins, McConville, and Propp proved that labeled chip-firing on a line always leaves the chips in sorted order if the number of chips is even. Here, we present a new proof of this result, based on a partial ordering of the firing moves near the end of the process, leading to temporary local confluence of the system. We then use the methods from this proof to resolve a series of previously open problems. Finally, we discuss our methods in relation to the general problem of which systems display various confluence properties.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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