EWtableaux, permutations and recurrent configurations of the sandpile model on Ferrers graphs
Seminar/Forum
The Abelian sandpile model (ASM) is a dynamic process on a graph. More specifically, it is a Markov chain on the set of configurations on that graph. Of particular interest are the recurrent configurations, i.e. those that appear infinitely often in the longtime running of the model. We study the ASM on Ferrers graphs, a class of bipartite graphs in onetoone correspondence with Ferrers diagrams. We show that minimal recurrent configurations are in onetoone correspondence with a set of certain 0/1 fillings of the Ferrers diagrams introduced by Ehrenborg and van Willigensburg. We refer to these fillings as EWtableaux, and establish a bijection between the set of EWtableaux of a given Ferrers diagram and a set of permutations whose descent bottoms are given by the shape of the Ferrers diagram. This induces a bijection between these permutations and minimal recurrent configurations of the ASM. We enrich this bijection to encode all recurrent configurations, via a decoration of the corresponding permutation. We also show that the set of recurrent configurations over all Ferrers graphs of a given size are in bijection with the set of alternating (or local binary search) trees of that size.
Presenter

Thomas Selig, University of Strathclyde