Repetitions of Pak-Stanley Labels in $G$-Shi Arrangements

Given a simple graph $G$, one can define a hyperplane arrangement called the $G$-Shi arrangement. The Pak-Stanley algorithm labels the regions of this arrangement with $G_\bullet$-parking functions. When $G$ is a complete graph, we recover the Shi arrangement, and the Pak-Stanley labels give a bijection with ordinary parking functions. However, for proper subgraphs $G \subset K_n$, while the Pak-Stanley labels still include every $G_{\bullet}$-parking function, some appear more than once. These repetitions of Pak-Stanley labels are a topic of interest in the study of $G$-Shi arrangements and $G_{\bullet}$-parking functions. Furthermore, $G_{\bullet}$-parking functions are connected to many other combinatorial objects (for example, superstable configurations in chip-firing). In studying these repetitions, we can draw on existing results about these objects such as Dhar's Burning Algorithm. Conversely, our results have implications for the study of these objects as well. The key insight of our work is the introduction of a combinatorial model called the Three Rows Game. Analyzing the histories of this game and how they induce identical outcomes lets us characterize the multiplicities of the Pak-Stanley labels. Using this model, we develop a classification theorem for the multiplicities of the Pak-Stanley labels of the regions in the $P_n$-Shi arrangement, where $P_n$ is the path graph on $n$ vertices. Then, we generalize the Three Rows Game into the $T$-Three Rows Game. This allows us to study the multiplicities of the Pak-Stanley labels of the regions in $T$-Shi arrangements, where $T$ is any tree. Finally, we discuss the possibilities and difficulties in applying our method to arbitrary graphs. In particular, we analyze multiplicities in the case when $G$ is a cycle graph, and prove a uniqueness result for maximal $G_{\bullet}$-parking functions for all graphs using the Three Rows Game.