On Ternary Trees and Fighting Fish
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916963288088576 |
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| author | Eu, Sen-Peng Fu, Tung-Shan Pan, Yu-Jen |
| author_facet | Eu, Sen-Peng Fu, Tung-Shan Pan, Yu-Jen |
| contents | Fighting fish is a combinatorial configuration introduced by Duchi et al. as a new model of branching surfaces that generalizes directed convex polyominoes. We come up with an alternative construction of fighting fish, using a tree structure built on the so-called stem cells of fighting fish. From this perspective, we establish a bijection between ternary trees and fighting fish with a marked strip of cells, which specializes to a direct bijection between left ternary trees and fighting fish. Using these results, we obtain a combinatorial enumeration of the fighting fish of size $n$ by establishing a $(n+1)$-to-2 bijection with the ternary trees having $n$ nodes. We present some additional enumerative results including that the fighting fish with a marked tail and the horizontally symmetric fighting fish are equinumerous with the ordered pairs of ternary trees having a total of a prescribed number of nodes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16667 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Ternary Trees and Fighting Fish Eu, Sen-Peng Fu, Tung-Shan Pan, Yu-Jen Combinatorics 05A19, 05A15 Fighting fish is a combinatorial configuration introduced by Duchi et al. as a new model of branching surfaces that generalizes directed convex polyominoes. We come up with an alternative construction of fighting fish, using a tree structure built on the so-called stem cells of fighting fish. From this perspective, we establish a bijection between ternary trees and fighting fish with a marked strip of cells, which specializes to a direct bijection between left ternary trees and fighting fish. Using these results, we obtain a combinatorial enumeration of the fighting fish of size $n$ by establishing a $(n+1)$-to-2 bijection with the ternary trees having $n$ nodes. We present some additional enumerative results including that the fighting fish with a marked tail and the horizontally symmetric fighting fish are equinumerous with the ordered pairs of ternary trees having a total of a prescribed number of nodes. |
| title | On Ternary Trees and Fighting Fish |
| topic | Combinatorics 05A19, 05A15 |
| url | https://arxiv.org/abs/2509.16667 |