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arXiv · 2401.06488

Shuffle theorems and sandpiles

Michele D'Adderio·Mark Dukes·Alessandro Iraci·Alexander Lazar·Yvan Le Borgne·Anna Vanden Wyngaerd
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Abstract

We provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs $\widehat{G}_{μ,ν}$, 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 $\langle \nabla e_n, e_μh_ν\rangle$ in terms of these configurations.

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Michele D'Adderio, Mark Dukes, Alessandro Iraci, Alexander Lazar, Yvan Le Borgne, Anna Vanden Wyngaerd. 2024-01-15. Shuffle theorems and sandpiles. https://arxiv.org/abs/2401.06488

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