The Abelian sandpile model on Ferrers graphs — a classification of recurrent configurations

Dukes, Mark and Selig, Thomas and Smith, Jason P. and Steingrímsson, Einar (2019) The Abelian sandpile model on Ferrers graphs — a classification of recurrent configurations. European Journal of Combinatorics, 81. pp. 221-241. ISSN 0195-6698 (https://doi.org/10.1016/j.ejc.2019.05.008)

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Abstract

We classify all recurrent configurations of the Abelian sandpile model (ASM) on Ferrers graphs. The classification is in terms of decorations of EW-tableaux, which undecorated are in bijection with the minimal recurrent configurations. We introduce decorated permutations, extending to decorated EW-tableaux a bijection between such tableaux and permutations, giving a direct bijection between the decorated permutations and all recurrent configurations of the ASM. We also describe a bijection between the decorated permutations and the intransitive trees of Postnikov, the breadth-first search of which corresponds to a canonical toppling of the corresponding configurations.

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