A Loehr-Remmel bijection in the $n\times kn$ grid and sandpiles

Abstract

We extend the $\mathsf{pmaj}$ statistic of Loehr and Remmel to labelled Dyck paths in the $n\times kn$ grid, and generalize their bijection sending the bistatistic $(\mathsf{dinv},\mathsf{area})$ to $(\mathsf{area},\mathsf{pmaj})$, proving in this way a new combinatorial formula for ${\nabla }^k{e}_n$ ($k\ge 1$). At $k=1$ we recover the original statistic and the original bijection. Moreover, we provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs $G_{\mu ,\nu }^{(k)}$, indexed by an integer $k\ge 1$ and two compositions $\mu$ and $\nu$: at $k=1$ these are the clique-independent graphs of D'Adderio et al. Finally, we define a $\mathsf{delay}$ 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 polynomials $⟨{\nabla }^k{e}_n,e_{\mu }{h}_{\nu }⟩$ from the $(n,kn)$-shuffle theorem. At $k=1$ we recover the main results of D'Adderio et al.

34 pages, 11 figures, 5 tables