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

Abstract

A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the tiered trees introduced by Dugan et al. [10]. This bijection allows certain parameters of the recurrent configurations to be read on the corresponding tree. In particular, we show that the level of a recurrent configuration can be interpreted as the external activity of the corresponding tree, so that the bijection exhibited provides a new proof of a famous result linking the level polynomial of the ASM to the ubiquitous Tutte polynomial. 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. Date

Tiered trees and Theta operators

“…Remark 7.9. In [12], the authors also provide a bijection between tiered trees and recurrent configurations on a permutation graph. They define an order on the edges of the graph such that the level of the configuration corresponds to the exterior activity of the tree.…”

“…Example 7.12. For P the path in Figure 9, we have The canonical toppling order of α(P ) is (8,3,12,9,10,11,5,4,1,7,2,6).…”

“…They are a generalisation of intransitive trees [26], which are tiered trees with only two levels. The notion is related to spanning trees of inversion graphs (Remark 3.14) and has connections to the abelian sandpile model on such graphs [12]. We slightly extend the definition of tiered trees to allow for non-distinct labels (see Definition 3.8).…”

Combinatorial aspects of sandpile models on wheel and fan graphs

“…• complete multi-partite graphs with a dominating sink [17], correspondence to so-called (p 1 , • • • , p k )-parking functions; • complete bipartite graphs where the sink is in one of the two components [24], correspondence to (labelled) paralllelogram polyominoes (see also [1,2,19]); • complete split graphs [23], correspondences to the so-called tiered parking functions and Motzkin words; • Ferrers graphs [26,39], correspondence to (decorated) EW-tableaux (see Section 3.2 for more details); • permutation graphs [25], correspondences to tiered trees and non-ambiguous binary trees. In this paper, we study combinatorial aspects (in the above sense) of the sandpile model on wheel and fan graphs.…”

Parallelogram polyominoes and rectangular EW-tableaux: Correspondences through the Sandpile model

“…In a series of recent papers, several classes of combinatorial objects have been shown to be in one-to-one correspondence with recurrent configurations of the Abelian sandpile model (ASM) on different classes of graphs [1,2,9,[13][14][15]. These connections have shown a certain combinatorial richness emerges from studying the behaviour of Dhar's burning algorithm -an algorithm that checks a configuration for recurrence -on different graph classes.…”