Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma

TL;DR

This paper introduces two operators on sandpile configurations that establish a bijection between recurrent and parking configurations, generalizes the Cyclic Lemma using bi-infinite paths, and enumerates classes of polyominoes.

Contribution

It presents a novel bijection on sandpile configurations, extends the Cyclic Lemma to new settings, and connects these to polyomino enumeration.

Findings

Operators provide a bijection preserving sandpile group classes

Generalization of the Cyclic Lemma via bi-infinite paths

Enumeration of classes of polyominoes

Abstract

We introduce two operators on stable configurations of the sandpile model that provide an algorithmic bijection between recurrent and parking configurations. This bijection preserves their equivalence classes with respect to the sandpile group. The study of these operators in the special case of the complete bipartite graph ${K}_{m,n}$ naturally leads to a generalization of the well known Cyclic Lemma of Dvoretsky and Motzkin, via pairs of periodic bi-infinite paths in the plane having slightly different slopes. We achieve our results by interpreting the action of these operators as an action on a point in the grid $\mathbb{Z}^2$ which is pointed to by one of these pairs of paths. Our Cyclic lemma allows us to enumerate several classes of polyominoes, and therefore builds on the work of Irving and Rattan (2009), Chapman et al. (2009), and Bonin et al. (2003).