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

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.

Permutation Graphs and the Abelian Sandpile Model, Tiered Trees and Non-Ambiguous Binary Trees

“…We show that the set of minimal recurrent configurations is in bijection with the set of complete non-ambiguous binary trees introduced by Aval et al [2], and introduce a multi-rooted generalization of these that we show to correspond to all recurrent configurations. In the case of permutations with a single descent, we recover some results from the case of Ferrers graphs presented in [11], while we also recover results of Perkinson et al [16] in the case of threshold graphs.…”

“…In this case the spanning trees of the permutation graph are the intransitive trees introduced by Postnikov [17]. As such, we recover results from [11,Section 5.3].…”

“…This paper generalizes the results in [11], where the recurrent configurations on Ferrers graphs were classified in terms of decorated EW-tableaux, since Ferrers graphs are isomorphic to permutation graphs of permutations with a single descent We extend the bijection in [11] between recurrent configurations on Ferrers graphs and the intransitive trees of Postnikov [17], to bijectively connect recurrent configurations of permutation graphs and the tiered trees introduced by Dugan et al [10], of which the intransitive trees are a special case.…”

“…where p(i) is the parent of i in the rooted tree T , and (p(i), i) is the interval [p(i) + 1, i − 1] if p(i) < i, and (p(i), i) = [i + 1, p(i) − 1] if i < p(i). In particular, we recover the bijection between the set of intransitive trees on n vertices and the set of recurrent configurations on Ferrers graphs on n vertices from [11].…”

“…In Section 4 we recall the definition of complete non-ambiguous binary trees introduced by Aval et al [2], show that these are in bijection with the set of minimal recurrent configurations of the ASM and introduce a generalization that we show to correspond to all recurrent configurations. Finally, in Section 5 we study two special cases of permutation graphs, namely Ferrers graphs (corresponding to permutations with a single descent) and threshold graphs, and recover results from [11] and [16] respectively.…”