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Permutation-based Strategies for Labeled Chip-Firing on$k$ -ary Trees
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Permutation-based Strategies for Labeled Chip-Firing on$k$ -ary Trees
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Journal Article
Permutation-based Strategies for Labeled Chip-Firing on$k$ -ary Trees
2026
Overview
Chip-firing is a combinatorial game on a graph, in which chips are placed and dispersed among its vertices until a stable configuration is achieved. We specifically study a chip-firing variant 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. When a vertex fires, we select$k$labeled chips and send the$i$ th smallest chip among them to its$i$ th leftmost child. A stable configuration is reached when no vertex can fire. In this paper, we focus on stable configurations resulting from specific firing strategies based on permutations of$1, 2, \\dots, n$ . We then express the stable configuration as a permutation of$0,1, 2, \\dots, k^n-1$and explore its properties, such as the number of inversions and descents. 20 pages, 4 figures, 1 table, v4: final version
Publisher
DMTCS
Subject
