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

Abstract

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 Km,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 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.

EW-Tableaux, Le-Tableaux, Tree-Like Tableaux and the Abelian Sandpile Model

“…There is a one-to-one correspondence between these decorated objects and all recurrent configurations of the Abelian sandpile model on Ferrers graphs. This complements and extends the work of Dukes and Le Borgne [11] (see also [3]) on the sandpile model on the complete bipartite graph, which is the special case of Ferrers graphs corresponding to rectangular Ferrers diagrams.…”

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

“…These graphs are a class of bipartite graphs which include the complete bipartite graphs (corresponding to rectangular Ferrers diagrams). As such, this work can be seen as complementing and extending that of [1,2,11,16].…”

“…Let T be the tableau from Example 3.2 so that φ T C (T ) = (0, 0, 1, 2, 1, 1, 0, 3). For ease of reference, let us note that the array recording the least unstable height of each vertex is (1,1,3,3,3,2,4,4). Also, let us underline those parts that correspond to the same part of the graph as the sink.…”

“…Example 3.11. Let c = (0, 0, 2, 1, 0, 0, 3, 2) and let F be the Ferrers diagram (5,3,3,2). The canonical toppling for this configuration is…”

“…j to be the number of neighbors of j in block V (p) , that is, d (p) j := |{ℓ ∈ V (p) : j < ℓ}|. By definition of the canonical toppling, in order for the vertex j to be in the block U (i) , the vertex j must be stable after toppling all vertices of V (1) , V (2) , . .…”