The Abelian sandpile model on Ferrers graphs -- A classification of recurrent configurations

TL;DR

This paper provides a comprehensive classification of all recurrent configurations of the Abelian sandpile model on Ferrers graphs using decorated permutations and EW-tableaux, establishing new bijections and combinatorial structures.

Contribution

It introduces decorated permutations and extends bijections between EW-tableaux, permutations, and recurrent configurations, offering a novel combinatorial framework for the ASM on Ferrers graphs.

Findings

Classified all recurrent configurations via decorated EW-tableaux.

Established a bijection between decorated permutations and recurrent configurations.

Connected the configurations to intransitive trees and canonical topplings.

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.