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 $\nabla^k e_n$ ($k \geq 1$). 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 $G_{μ, ν}^{(k)}$, indexed by an integer $k \geq 1$ and two compositions $μ$ and $ν$: 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 $\langle \nabla^k e_n,e_μh_ν\rangle$ from the $(n,kn)$-shuffle theorem. At $k = 1$ we recover the main results of D'Adderio et al.