Abelian sandpile model and Biggs-Merino polynomial for directed graphs

We prove several results concerning a polynomial that arises from the sandpile model on directed graphs; these results are previously only known for undirected graphs. Implicit in the sandpile model is the choice of a sink vertex, and it was conjectured by Perrot and Pham that the polynomial c0+c1y+…cnyn, where ci is the number of recurrent classes of the sandpile model with level i, is independent of the choice of the sink. We prove their conjecture by expressing the polynomial as an invariant of the sinkless sandpile model. We then present a bijection between arborescences of directed graphs and reverse G-parking functions that preserves external activity by generalizing Cori-Le Borgne bijection for undirected graphs. As an application of this bijection, we extend Merino's Theorem by showing that for any Eulerian directed graph the polynomial c0+c1y+…cnyn is equal to the greedoid polynomial of the graph.