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The sandpile model on the complete split graph: $q,t$-Schröder polynomials, sawtooth polyominoes, and a cycle lemma

Henri Derycke, Mark Dukes, Yvan Le Borgne

TL;DR

The paper analyzes the Abelian sandpile model on the complete split graph $S_{n,d}$ by introducing two parallel toppling schemes, CTI and ITC, and associated generating polynomials that encode height/level and delay statistics. It builds a bridge from sorted recurrent configurations to Schröder paths and, further, to sawtooth polyominoes, with CTI/ITC topplings corresponding to bounce paths on these polyominoes. A key result is the ITC–Schröder connection, which yields a symmetry of the $q,t$-polynomial and ties to symmetric function theory; a cycle lemma then provides a clean enumeration of sorted recurrent configurations. The work also presents sawtooth polyomino representations that visualize toppling dynamics and offers conjectures about the equivalence of CTI and ITC generating functions, suggesting a deeper combinatorial structure behind sandpile configurations on split graphs.

Abstract

This paper studies sorted recurrent configurations of the Abelian sandpile model on the complete split graph. We introduce two natural toppling processes, CTI and ITC toppling, on the recurrent configurations and use these to define two toppling delay statistics, wtopple$_{CTI}$ and wtopple$_{ITC}$. These new toppling delay statistics are time-weighted sums for the number of vertices that topple during each iteration of the toppling processes. We then introduce the bivariate $q,t$-CTI and $q,t$-ITC polynomials that are the generating functions of the bistatistics (level,wtopple$_{ITC}$) and (level,wtopple$_{CTI}$), where level is the well-established sandpile level statistic. We prove the bistatistic (level,wtopple$_{ITC}$) maps to a bistatistic (area,bounce) on Schröder paths that was introduced by Egge, Haglund, Killpatrick and Kremer (2003). This establishes equality of the $q,t$-ITC polynomial and the $q,t$-Schröder polynomial of those same authors. This connection allows us to relate the $q,t$-ITC polynomial to the theory of symmetric functions and also establishes symmetry of the $q,t$-ITC polynomials. We conjecture equality of the $q,t$-CTI and $q,t$-ITC polynomials. We also present and prove a characterization of sorted recurrent configurations as a new class of polyominoes that we call sawtooth polyominoes. The CTI and ITC toppling processes on sorted recurrent configurations are proven to correspond to bounce paths within the polyominoes. The main difference between the two bounce paths is the initial direction in which they travel. In addition to this, and building on the results of Aval, D'Adderio, Dukes, and Le Borgne (2016), we present a cycle lemma for a slight extension of stable configurations that allows for an enumeration of sorted recurrent configurations within the framework of the sandpile model.

The sandpile model on the complete split graph: $q,t$-Schröder polynomials, sawtooth polyominoes, and a cycle lemma

TL;DR

The paper analyzes the Abelian sandpile model on the complete split graph by introducing two parallel toppling schemes, CTI and ITC, and associated generating polynomials that encode height/level and delay statistics. It builds a bridge from sorted recurrent configurations to Schröder paths and, further, to sawtooth polyominoes, with CTI/ITC topplings corresponding to bounce paths on these polyominoes. A key result is the ITC–Schröder connection, which yields a symmetry of the -polynomial and ties to symmetric function theory; a cycle lemma then provides a clean enumeration of sorted recurrent configurations. The work also presents sawtooth polyomino representations that visualize toppling dynamics and offers conjectures about the equivalence of CTI and ITC generating functions, suggesting a deeper combinatorial structure behind sandpile configurations on split graphs.

Abstract

This paper studies sorted recurrent configurations of the Abelian sandpile model on the complete split graph. We introduce two natural toppling processes, CTI and ITC toppling, on the recurrent configurations and use these to define two toppling delay statistics, wtopple and wtopple. These new toppling delay statistics are time-weighted sums for the number of vertices that topple during each iteration of the toppling processes. We then introduce the bivariate -CTI and -ITC polynomials that are the generating functions of the bistatistics (level,wtopple) and (level,wtopple), where level is the well-established sandpile level statistic. We prove the bistatistic (level,wtopple) maps to a bistatistic (area,bounce) on Schröder paths that was introduced by Egge, Haglund, Killpatrick and Kremer (2003). This establishes equality of the -ITC polynomial and the -Schröder polynomial of those same authors. This connection allows us to relate the -ITC polynomial to the theory of symmetric functions and also establishes symmetry of the -ITC polynomials. We conjecture equality of the -CTI and -ITC polynomials. We also present and prove a characterization of sorted recurrent configurations as a new class of polyominoes that we call sawtooth polyominoes. The CTI and ITC toppling processes on sorted recurrent configurations are proven to correspond to bounce paths within the polyominoes. The main difference between the two bounce paths is the initial direction in which they travel. In addition to this, and building on the results of Aval, D'Adderio, Dukes, and Le Borgne (2016), we present a cycle lemma for a slight extension of stable configurations that allows for an enumeration of sorted recurrent configurations within the framework of the sandpile model.
Paper Structure (17 sections, 21 theorems, 116 equations, 11 figures, 1 table)

This paper contains 17 sections, 21 theorems, 116 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. The sandpile model on $S_{n,d}$ and two toppling processes
  3. CTI toppling
  4. ITC toppling
  5. Schröder paths and ITC-topplings
  6. Mapping configuration height to Schröder path area
  7. Mapping $\mathsf{wtopple}_{ITC}$ to the statistic $\mathsf{bounce}^{\mathsf{Sch}}$
  8. Sorted recurrent configurations as sawtooth polyominoes
  9. Sawtooth polyominoes and Schröder paths
  10. Mapping sorted recurrent configurations to sawtooth polyominoes
  11. CTI topplings and bounce paths within sawtooth polyominoes
  12. ITC topplings and bounce paths within sawtooth polyominoes
  13. A cycle lemma to count sorted recurrent configurations on split graphs
  14. Conclusion
  15. A proof of Proposition \ref{['prop:countingFqt']}
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $G$ be a graph with sink $s$, and let $c$ be a stable configuration on $G$. Then $c$ is recurrent if and only if there exists an ordering $v_0=s,v_1,\ldots,v_{n}$ of the vertices of $G$ such that, starting from $c$, for any $i \geq 1$, toppling the vertices $v_0,\ldots,v_{i-1}$ causes the vertex

Figures (11)

  • Figure 1: The complete split graph $S_{5,3}$. The sink is in the dashed rectangle, the 5 vertices below it form the clique part, and the $3$ vertically aligned vertices to its right are the independent part.
  • Figure 2: The Schröder path $w= UHUDUHHDUDUDD\in \mathsf{Schr\ddot{o}der}_{5,3}$ from Example \ref{['example:three:two']} is illustrated by the blue line in the left diagram. The Dyck path $C(w)$ that represents the collapse of this path, achieved by removing horizontal steps, is illustrated to the right using a blue line. The bounce path of $C(w)$ starts at $(5,5)$ and runs to $(0,0)$. Since it hits the diagonal at positions $(3,3)$ and $(1,1)$ we have $\mathrm{bounce}(C(w)) = 1+3=4$. The peaks of the bounce path are illustrated with red dots in the right diagram. These peaks are copied to the tops of the corresponding $U$ steps on the original Schröder path. The sum $\sum_{\alpha} b(\alpha)$ is the sum over all $H$ steps $\alpha$ in the Schröder path of the statistic $b(\alpha)$, which represents the number of red dots above $\alpha$ in the diagram.
  • Figure 3: The Schröder path $w= UUDUDUHHDUDHD\in \mathsf{Schr\ddot{o}der}_{5,3}$ from Example \ref{['example:three:four']}.
  • Figure 4: Configuration $c=(7_C,7_C,6_C,5_C,2_C;3_I,3_I,1_I) \in \mathsf{SortedRec}(S_{5,3})$ and its related Schröder word $w=UHUDUHHDUDUDD$ that corresponds to the example Schröder path in Haglund haglund-qtschroder. The peaks are indicated with red dots and the bounce path is the red dotted line. To the right we follow the ITC-toppling process for $c$. Note that the degree of a clique vertex is $n+d=8$ while the degree of an independent vertex is $n+1=6$. The green, magenta, and orange triangles are explained in the proof of Theorem \ref{['lem:itc-area']}. Green triangles correspond to clique vertices/columns, while the other two colours correspond to independent vertices/columns. A triangle associated with an independent vertex will be orange if there are horizontal $H$ steps on the path in both its column and row. Otherwise those triangles are magenta.
  • Figure 5: A word having prefix $H$. Here $w=HHHw'$ so that $j=3$.
  • ...and 6 more figures

Theorems & Definitions (67)

  • Proposition 2.1: Dhar, Section 6.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Theorem 2.7
  • proof
  • Example 2.8
  • Definition 3.1
  • ...and 57 more