{
  "abstract": "We investigate counting problems for discrete (geometric) structures in tropical settings, motivated by the comparison of the expressivity of linear programs and mean-payoff games. First, we give an upper bound on the number of coherent matching fields of type $(n,r)$ and a lower bound on the number of combinatorial types of $n$-hyperplane sequences in the max-plus tropical space of dimension $r-1$. These structures are closely related, and our bounds establish that as $n \\to \\infty$, for every fixed $r$ their number equals $n^{(r-1)^2n+o(n)}$. This in turn implies that the number of regular triangulations of the products of simplices $Δ^{n-1}\\timesΔ^{r-1}$ is also $n^{(r-1)^2n+o(n)}$, thereby answering a question of Santos. Second, we obtain that the number of labeled order types over the reals and over the signed tropical numbers coincide at first order, establishing a comparison of the expressivity of mean-payoff games and linear programs as measured by labeled order types. Last, the subclass of mean-payoff games known as parity games arise from linear programs by a ``double tropicalization.'' We show that the number of labeled order types obtained after this double tropicalization collapses.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40622",
  "author_total": 4,
  "authors": [
    {
      "name": "Xavier Allamigeon"
    },
    {
      "name": "Yiyuan Chen"
    },
    {
      "name": "Stéphane Gaubert"
    },
    {
      "name": "Xavier Goaoc"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6yj01hsnr1cec.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40622",
  "fulltext_url": null,
  "id": "6yj01hsnr1cec",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40622",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40622",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6yj01hsnr1cec.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:35Z",
      "native_id": "2610.40622",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40622"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Number of Tropical Order Types and Coherent Matching Fields",
  "title_text": "The Number of Tropical Order Types and Coherent Matching Fields",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:53Z",
  "url": "https://research.draconitic.com/paper/6yj01hsnr1cec",
  "venue": "arXiv",
  "versions": []
}