Labeled Chip-Firing on Star Graphs

Chip-firing is a combinatorial game played on a graph in
which ?chips? are placed on the vertices and redistributed according
to a fixed rule. In 2016, Hopkins, McConville, and Propp introduced a
labeled variant of this game, where each chip carries a distinct
label. They proved an interesting ?sorting? property for the game on
the infinite path graph; however, their proof is long and technically
involved. In 2020, Klivans and Liscio gave a significantly simpler
proof using a structure they call the?<em>firing order poset</em>.
This tool enables deeper analysis of the game on broader classes of
graphs, and recent work has explored generalizations to binary
and?k-ary trees.?

In this talk, I will provide a brief overview of
these ideas before presenting a generalization of the labeled
chip-firing game to infinitely subdivided k-star graphs, which exhibit
a similar sorting property and an intriguing connection to standard
Young tableaux. This is based on work with Ryan Lynch and Annika
Gonzalez-Zugasti.