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.
- COMPLETE BIPARTITE GRAPH
- PARALLELOGRAM POLYOMINOES
Dukes, M., Selig, T., Smith, J. P., & Steingrimsson, E. (2019). The Abelian sandpile model on Ferrers graphs - A classification of recurrent configurations. European Journal of Combinatorics, 81, 221-241. https://doi.org/10.1016/j.ejc.2019.05.008