The bottom-triangle row-count conjecture for chip-firing configurations

A configuration F(x,y)F(x,y) consists of rows of chips, with the bottom triangle formed by the minimal rows at the bottom of the configuration. Let the longest row be a row having maximal length among all rows in the configuration. Bottom-triangle row-count conjecture. The number of rows in the bottom triangle equals the length of the longest row minus 11. This is motivated by computations of F(x,y)F(x,y) for nn up to and including 1212, which show that the intermediate configuration ends with a long rectangle followed by the bottom triangle, with the rectangle as its widest part. The claim remains conjectural beyond these computations.

Sources & referencesView supporting material

Primary source

Ryota Inagaki, Tanya Khovanova and Austin Luo, “Chip-firing on the Lattice of Nonnegative Integer Points”, arXiv:2601.09125 (2026).

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