Shuffle theorems and sandpiles

2024/01/12 by Michele D’Adderio, D'Adderio, Michele, Mark Dukes +8 · 1 citation
Physics and Astronomy · Mathematics · Computer Science · #Theoretical and Computational Physics #Advanced Combinatorial Mathematics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2401.06488

Abstract

We provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs \widehatG_μ,ν, which we call clique-independent graphs, indexed by two compositions μ and ν. 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 ⟨ ∇ e_n, e_μh_ν⟩ in terms of these configurations.

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