Shuffle theorems and sandpiles

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 $⟨\mathrm{\nabla }{e}_n,e_{\mu }{h}_{\nu }⟩$ in terms of these configurations.