Counting Spinal Tree-Child Networks via Word Encodings and Generating Functions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866913112909676544 |
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| author | Vives, Pau de Mier, Anna Cardona, Gabriel Pons, Joan Carles |
| author_facet | Vives, Pau de Mier, Anna Cardona, Gabriel Pons, Joan Carles |
| contents | We study the enumeration of spinal tree-child phylogenetic networks, a rigid family of tree-child networks in which all internal vertices lie on a single root--to--leaf path. We provide two complementary combinatorial frameworks. First, we introduce a word model: unlabeled spinal networks correspond to a suitable class of restricted words with fixed multiplicities, taken modulo a simple relabeling equivalence, which yields an explicit closed enumeration. Second, we develop a symbolic-method approach based on a marked version of trees that admits a clean recursive specification; its boxed-product translation leads to a solvable bivariate generating function and a direct derivation of the coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10926 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting Spinal Tree-Child Networks via Word Encodings and Generating Functions Vives, Pau de Mier, Anna Cardona, Gabriel Pons, Joan Carles Combinatorics We study the enumeration of spinal tree-child phylogenetic networks, a rigid family of tree-child networks in which all internal vertices lie on a single root--to--leaf path. We provide two complementary combinatorial frameworks. First, we introduce a word model: unlabeled spinal networks correspond to a suitable class of restricted words with fixed multiplicities, taken modulo a simple relabeling equivalence, which yields an explicit closed enumeration. Second, we develop a symbolic-method approach based on a marked version of trees that admits a clean recursive specification; its boxed-product translation leads to a solvable bivariate generating function and a direct derivation of the coefficients. |
| title | Counting Spinal Tree-Child Networks via Word Encodings and Generating Functions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.10926 |