Permutation graphs and the Abelian sandpile model, tiered trees and non-ambiguous binary trees

TL;DR

This paper explores the Abelian sandpile model on permutation graphs, establishing bijections with tiered trees and non-ambiguous binary trees, linking sandpile configurations to well-studied combinatorial structures and classical graph invariants.

Contribution

It introduces new bijections between recurrent configurations of the ASM on permutation graphs and combinatorial trees, providing novel proofs and generalizations for known graph invariants.

Findings

Recurrent configurations correspond to tiered trees with parameters read as tree properties.

Minimal recurrent configurations biject with complete non-ambiguous binary trees.

Results recover known cases for Ferrers and threshold graphs.

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.