Labeled Chip-Firing on Directed $k$-ary Trees and Where Chips Land
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916810971938816 |
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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 played on a graph, in which chips are placed and dispersed on the vertices until a stable configuration is achieved. We study a chip-firing variant on an infinite, rooted directed $k$-ary tree, where we place $k^n$ chips labeled $1,2,3,\dots, k^n$ on the root for some nonnegative integer $n$. A vertex $v$ can fire if it has at least $k$ chips; when it fires, $k$ chips are selected, and the chip with the $i$th smallest label is sent to the $i$th leftmost child of $v$. A stable configuration is reached when no vertices can fire. In this paper, we prove numerous properties of the stable configuration, such as that chips land on vertices in ranges and the lengths of those ranges. We also describe where each chip can land. This helps us describe possible stable configurations of the game. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Labeled Chip-Firing on Directed $k$-ary Trees and Where Chips Land Inagaki, Ryota Khovanova, Tanya Luo, Austin Combinatorics 05C57, 05C63, 05A05 Chip-firing is a combinatorial game played on a graph, in which chips are placed and dispersed on the vertices until a stable configuration is achieved. We study a chip-firing variant on an infinite, rooted directed $k$-ary tree, where we place $k^n$ chips labeled $1,2,3,\dots, k^n$ on the root for some nonnegative integer $n$. A vertex $v$ can fire if it has at least $k$ chips; when it fires, $k$ chips are selected, and the chip with the $i$th smallest label is sent to the $i$th leftmost child of $v$. A stable configuration is reached when no vertices can fire. In this paper, we prove numerous properties of the stable configuration, such as that chips land on vertices in ranges and the lengths of those ranges. We also describe where each chip can land. This helps us describe possible stable configurations of the game. |
| title | Labeled Chip-Firing on Directed $k$-ary Trees and Where Chips Land |
| topic | Combinatorics 05C57, 05C63, 05A05 |
| url | https://arxiv.org/abs/2506.20656 |