Shuffle Theorems and Sandpiles

Abstract

We provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs \({\widehat{G}}_{\mu ,\nu }\), which we call clique-independent graphs, indexed by two compositions \(\mu \) and \(\nu \). Moreover, we define a delay statistic on these configurations, and we show that, together with the usual level statistic, it can be used to provide a new combinatorial interpretation of the celebrated shuffle theorem of Carlsson and Mellit. More precisely, we will see how to interpret the polynomials \(\langle \nabla e_n, e_\mu h_\nu \rangle \) in terms of these configurations.