Labeled Chip-firing on Binary Trees with $2^n-1$ Chips

TL;DR

This paper investigates labeled chip-firing on binary trees, establishing a sorting property of terminal configurations and proving that the endgame moves form a modular lattice structure.

Contribution

It introduces new structural insights into labeled chip-firing on binary trees, including the sorting property and the lattice structure of endgame moves.

Findings

Terminal configurations exhibit a sorting property.

Endgame moves form a modular lattice.

Provides structural analysis of chip-firing dynamics.

Abstract

We study labeled chip-firing on binary trees and some of its modifications. We prove a sorting property of terminal configurations of the process. We also analyze the endgame moves poset and prove that this poset is a modular lattice.