Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Shuffle theorems and sandpiles

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

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.

Shuffle theorems and sandpiles

Abstract

We provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs , which we call clique-independent graphs, indexed by two compositions and . 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 in terms of these configurations.
Paper Structure (7 sections, 8 theorems, 39 equations, 2 figures, 1 algorithm)

This paper contains 7 sections, 8 theorems, 39 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Combinatorics of the shuffle theorem
  3. The clique-independent graphs Gmunu
  4. Basics of the sandpile model
  5. The toppling algorithm and the delay statistic
  6. Sorted recurrent configurations of Gmunu
  7. Bijection with parking functions

Key Result

Theorem 1.1

For every pair of compositions $\mu$, $\nu$ such that $n = \lvert \mu \rvert + \lvert \nu \rvert$ we have

Figures (2)

  • Figure 1: An element $D$ of $\mathsf{PF}((4,3);(3,2))$. The colors and crosses are explained in Example \ref{['ex:bijection']}.
  • Figure 2: The graph $\widehat{G}_{(4, \textcolor{olive}{3}), (\textcolor{red}{3}, \textcolor{blue}{2})}$. The vertices $\textcolor{red}{1}$, $\textcolor{red}{2}$ and $\textcolor{red}{3}$ have degree $10$, the vertices $\textcolor{blue}{4}$ and $\textcolor{blue}{5}$ have degree $11$, and all the other vertices have degree $12$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Theorem 2.9
  • ...and 30 more