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

Jean-Christophe Aval, Michele D'Adderio, Mark Dukes, Yvan Le Borgne

Research output: Contribution to journalArticle

6 Citations (Scopus)
18 Downloads (Pure)

Abstract

We introduce two operators on stable configurations of the sandpile model that provide an algorithmic bijection between recurrent and parking congurations. 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-in nite 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 Z2 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).
Original languageEnglish
Pages (from-to)59-98
Number of pages40
JournalAdvances in Applied Mathematics
Volume73
Early online date16 Nov 2015
Publication statusPublished - 1 Feb 2016

Keywords

  • sandpile model
  • complete graphs
  • complete bipartite graph
  • cyclic lemma

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