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Chip Firing on Directed $k$-ary Trees

Ryota Inagaki, Tanya Khovanova, Austin Luo

Abstract

Chip-firing is a combinatorial game played on a graph in which we place and disperse chips on vertices until a stable state is reached. We study a chip-firing variant played on an infinite rooted directed $k$-ary tree, where we place $k^\ell$ chips on the root for some positive integer $\ell$, and we say a vertex $v$ can fire if it has at least $k$ chips. A vertex fires by dispersing one chip to each out-neighbor. Once every vertex has less than $k$ chips, we reach a stable configuration since no vertex can fire. We determine the exact number and properties of the possible stable configurations of chips in the setting where chips are distinguishable.

Chip Firing on Directed $k$-ary Trees

Abstract

Chip-firing is a combinatorial game played on a graph in which we place and disperse chips on vertices until a stable state is reached. We study a chip-firing variant played on an infinite rooted directed -ary tree, where we place chips on the root for some positive integer , and we say a vertex can fire if it has at least chips. A vertex fires by dispersing one chip to each out-neighbor. Once every vertex has less than chips, we reach a stable configuration since no vertex can fire. We determine the exact number and properties of the possible stable configurations of chips in the setting where chips are distinguishable.

Paper Structure

This paper contains 15 sections, 14 theorems, 13 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Unlabeled Chip-Firing on Directed Graphs
  3. Labeled Chip-Firing on Directed Graphs
  4. Objectives and Roadmap
  5. Definitions and Basic Results
  6. Definitions
  7. Unlabeled Chip-Firing on Directed k-ary Trees
  8. Labeled Chip-Firing on Directed k-ary Trees
  9. Counting the Number of Stable Configurations
  10. Sequences that Can Appear in the Stable Configuration
  11. The Digit-Reversal Permutation
  12. Number of Inversions
  13. The Longest Decreasing Subsequence in Zkell
  14. Decreasing Sequences in Stable Configurations
  15. Acknowledgments

Key Result

Theorem 1.1

For a directed graph $G$ and initial configuration $c$ of chips on the graph, the unlabeled chip-firing game will either run forever or end after the same number of moves and at the same stable configuration. Furthermore, the number of times each vertex fires is the same regardless of the sequence o

Figures (6)

  • Figure 1: Example of unlabeled chip-firing on an infinite directed, rooted binary tree
  • Figure 2: Example of confluence breaking
  • Figure 3: Labeling for $2$ layers in directed $5$-ary tree
  • Figure 4: Example of labeled chip-firing in a directed binary tree with $4$ chips
  • Figure 5: Example of a walk of length $8$ in $\mathbb{R}^2$ with the diagonal $y =x$
  • ...and 1 more figures

Theorems & Definitions (52)

  • Example 1
  • Theorem 1.1: Theorem 1.1 MR1203679
  • Example 2
  • Example 3
  • Lemma 2.1
  • proof
  • Definition 1
  • Example 4
  • Lemma 3.1
  • proof
  • ...and 42 more