{
  "abstract": "Motivated by a word encoding of tree-child networks, we introduce trident tableaux, which are Young tableaux with a unique three-cell column satisfying certain conditions on successors. We construct a bijection between trident tableaux with $n+1$ columns and two-wall tableaux with $n$ columns, that is, two-row fillings with two designated columns in which vertical order is not imposed. The bijection matches the three-part decompositions of the two classes and shows that each class has cardinality $n(n+1)C_n/2$, where $C_n$ is the $n$-th Catalan number. In particular, it gives a trident-tableau interpretation of a shifted form of OEIS A002457. As a consequence, we obtain an exact enumeration of tree-child networks with one reticulation node that contain a trident, and show that their proportion among all tree-child networks with one reticulation node tends to $1/8$ as the number of leaves tends to infinity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28892",
  "author_total": 1,
  "authors": [
    {
      "name": "Hexuan Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fr4x9zmsmmmqf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28892",
  "fulltext_url": null,
  "id": "fr4x9zmsmmmqf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28892",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214392066",
  "pdf_url": "https://arxiv.org/pdf/2609.28892",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fr4x9zmsmmmqf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28892",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28892"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Trident Tableaux for Tree-Child Networks with One Reticulation Node: A Bijection with Two-Wall Tableaux",
  "title_text": "Trident Tableaux for Tree-Child Networks with One Reticulation Node: A Bijection with Two-Wall Tableaux",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/fr4x9zmsmmmqf",
  "venue": "arXiv",
  "versions": []
}