Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma
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 biinfinite 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).
 Publication:

arXiv eprints
 Pub Date:
 July 2013
 DOI:
 10.48550/arXiv.1307.7740
 arXiv:
 arXiv:1307.7740
 Bibcode:
 2013arXiv1307.7740A
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 28 pages