Permutation-based Strategies for Labeled Chip-Firing on $k$-ary Trees
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908761951567872 |
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| author | Inagaki, Ryota Khovanova, Tanya Luo, Austin |
| author_facet | Inagaki, Ryota Khovanova, Tanya Luo, Austin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Permutation-based Strategies for Labeled Chip-Firing on $k$-ary Trees Inagaki, Ryota Khovanova, Tanya Luo, Austin Combinatorics 05C57, 05C63, 05A05, 05A15 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. |
| title | Permutation-based Strategies for Labeled Chip-Firing on $k$-ary Trees |
| topic | Combinatorics 05C57, 05C63, 05A05, 05A15 |
| url | https://arxiv.org/abs/2503.09577 |