{
  "abstract": "We calculate the asymptotics of the number of connected, labeled, genus $g$ Eulerian maps with an arbitrary degree sequence, in the limit as the total number of vertices tends to infinity. This asymptotic is universal, in the sense that the leading-order term depends on only three map characteristics, regardless of the choice of the degree sequence. The constant factor in this formula is related to the Painlevé I equation. Our methods combine for the first time the analysis of the recurrence coefficients associated to a particular family of orthogonal polynomials, and the theory of analytic combinatorics of several variables. We also derive an exact formula for the number of connected, labeled, genus $1$ Eulerian maps. These are the first results on this kind of enumeration problem for $g\\geq 1$, non-regular (mixed-valence) maps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.14342",
  "author_total": 3,
  "authors": [
    {
      "name": "Ahmad Barhoumi"
    },
    {
      "name": "Roozbeh Gharakhloo"
    },
    {
      "name": "Nathan Hayford"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f01wcge9rbswj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.14342",
  "fulltext_url": null,
  "id": "f01wcge9rbswj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.14342",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7169504166",
  "pdf_url": "https://arxiv.org/pdf/2607.14342",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f01wcge9rbswj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:41Z",
      "native_id": "2607.14342",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.14342"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Universal Asymptotics and Exact Enumeration of Eulerian Maps",
  "title_text": "Universal Asymptotics and Exact Enumeration of Eulerian Maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/f01wcge9rbswj",
  "venue": "arXiv",
  "versions": []
}