Enumeration of weighted plane trees by a permutation model
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866909987735863296 |
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| author | Lu, Sicheng Song, Yi |
| author_facet | Lu, Sicheng Song, Yi |
| contents | This work addresses an enumeration problem on weighted bi-colored plane trees with prescribed vertex data, with all vertices labeled distinctly. We give a bijection proof of the enumeration formula originally due to Kochetkov, hence affirmatively answer a question of Adrianov-Pakovich-Zvonkin. The argument is purely combinatorial and totally constructive, remaining valid for real-valued edge weights. A central process is a geometric construction that directly encodes each tree as a permutation. We also exhibit algebraic relationships between the enumeration problem, the partial order on partitions of vertices and the Stirling numbers of the second kind. Some computation examples are presented as appendices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07544 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Enumeration of weighted plane trees by a permutation model Lu, Sicheng Song, Yi Combinatorics Primary 05C30 05A19, Secondary 05C10 57M15 This work addresses an enumeration problem on weighted bi-colored plane trees with prescribed vertex data, with all vertices labeled distinctly. We give a bijection proof of the enumeration formula originally due to Kochetkov, hence affirmatively answer a question of Adrianov-Pakovich-Zvonkin. The argument is purely combinatorial and totally constructive, remaining valid for real-valued edge weights. A central process is a geometric construction that directly encodes each tree as a permutation. We also exhibit algebraic relationships between the enumeration problem, the partial order on partitions of vertices and the Stirling numbers of the second kind. Some computation examples are presented as appendices. |
| title | Enumeration of weighted plane trees by a permutation model |
| topic | Combinatorics Primary 05C30 05A19, Secondary 05C10 57M15 |
| url | https://arxiv.org/abs/2601.07544 |