Permutation-based Strategies for Labeled Chip-Firing on $k$-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 configuration is reached. We study a chip-firing variant played on an infinite rooted directed $k$-ary tree, where we place $k^n$ chips labeled $0,1,\dots, k^n-1$ on the root for some nonnegative integer $n$, 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. In this paper, we focus on stable configurations resulting from applying a strategy $F_w$ corresponding to a permutation $w = w_1w_2\dots w_n\in S_n$: for each vertex $v$ on level $i$ of the $k$-ary tree, the chip with $j$ as its $w_i$th most significant digit in the $k$-ary expansion gets sent to the $(j+1)$st child of $v$. We express the stable configuration as a permutation, and we explore the properties of these permutations, such as the number of inversions, descents, and the descent set.