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Labeled Chip-Firing on Star Graphs

Annika Gonzalez-Zugasti, Ryan Lynch, Dylan Snustad

TL;DR

This paper analyzes labeled chip-firing on infinitely subdivided $k$-star graphs, focusing on configurations obtained when $km$ labeled chips start at the center. It introduces endgame fires and a book-like poset to prove a sorting property: in every reachable stable configuration the chips on each branch are ordered from center outward. For $m=2$, there is a bijection between reachable configurations and standard Young tableaux of shape $(2)^k$, counted by Catalan numbers; for general $m$, a volatility-minimizing firing rule yields a correspondence with SYT of shape $(m)^k$, though not all such tableaux are reachable. These results connect chip-firing dynamics on star graphs to classical combinatorics, providing structural and enumerative insights and pointing to rich avenues for further study, including extensions beyond SYT and probabilistic firing analyses.

Abstract

We study the stable configurations of the labeled chip-firing game on an infinitely subdivided $k$-star graph starting with $km$ chips on the center vertex. We prove a sorting property of this game and analyze special stable configurations corresponding to standard Young tableaux.

Labeled Chip-Firing on Star Graphs

TL;DR

This paper analyzes labeled chip-firing on infinitely subdivided -star graphs, focusing on configurations obtained when labeled chips start at the center. It introduces endgame fires and a book-like poset to prove a sorting property: in every reachable stable configuration the chips on each branch are ordered from center outward. For , there is a bijection between reachable configurations and standard Young tableaux of shape , counted by Catalan numbers; for general , a volatility-minimizing firing rule yields a correspondence with SYT of shape , though not all such tableaux are reachable. These results connect chip-firing dynamics on star graphs to classical combinatorics, providing structural and enumerative insights and pointing to rich avenues for further study, including extensions beyond SYT and probabilistic firing analyses.

Abstract

We study the stable configurations of the labeled chip-firing game on an infinitely subdivided -star graph starting with chips on the center vertex. We prove a sorting property of this game and analyze special stable configurations corresponding to standard Young tableaux.

Paper Structure

This paper contains 10 sections, 13 theorems, 15 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A Review of Unlabeled Chip-Firing
  3. Unlabeled Chip-Firing on the Infinite $k$-Star
  4. Labeled Chip-Firing on the Infinite $k$-Star
  5. Labeled Chip-Firing on the Infinite $k$-Star
  6. Main Theorem
  7. Categorization of Reachable Stable Configurations
  8. Shape $k\times 2$ SYT and Catalan Numbers
  9. Rectangular SYTs
  10. Future Questions

Key Result

Theorem 1.1

In labeled chip-firing on an infinite binary tree with $2^n-1$ chips (labeled from $1$ to $2^n-1$) initially placed at the origin, the terminal configuration always has one chip at each node of the first $n$ levels. Moreover, the bottom straight left and right descendants of a node contain the small

Figures (5)

  • Figure 1: $\operatorname{instar}(3)$ with vertices labeled.
  • Figure 2: An example of one page corresponding to the first branch of the book poset of $[\Delta^{k,5}]$ for some $k$. The red dotted lines indicate relations between elements of the endgame fires of the book poset and non-endgame fires, which we do not discuss in detail in our paper.
  • Figure 3: Counterexample to a generalized $\mathcal{T}^{k,m}\leftrightarrow \{$standard Young tableaux of shape $(m)^k\}$ bijection.
  • Figure 4: Tableaux of shape $(2)^4$ which are row-sorted and have first and last columns sorted but do not correspond to stable configurations reachable from $\Delta^{2,4}$. Non-increasing columns are highlighted in blue.
  • Figure 5: Stable configurations reachable from $\Delta^{k,m}$ for selected small $(k,m)$ pairs, as well as the number of stabilization sequences for each configuration. Configurations are read so that the first set of brackets indicates chips on branch 1 ordered from the center outwards, the second set of brackets indicates those on branch 2, and so on. Totally sorted configurations are highlighted in blue.

Theorems & Definitions (24)

  • Theorem 1.1: musiker2023labeledchipfiringbinarytrees
  • Theorem 2.1: BJORNER1991283
  • Corollary 3.1: Global confluence
  • Proposition 3.2
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 14 more