{
  "abstract": "Monge matrices arise throughout combinatorial optimization and algorithm design; the Monge polytope $\\mathcal{M}_{pq}$ is the set of $p \\times q$ Monge matrices lying inside the standard simplex on the set of matrix coordinates. We find a Stanley decomposition of the associated affine semigroup, and use it to obtain a volume formula for $\\mathcal{M}_{pq}$ expressed as a sum over \"Z-avoiding\" Delannoy paths in a $p \\times q$ grid. An efficient dynamic-programming implementation of this formula computes the volume in dimensions far beyond the reach of general-purpose exact-volume algorithms (e.g., the volume of $\\mathcal{M}_{20,20}$, which has dimension 399, is computed in a fraction of a second). As a corollary of our Stanley decomposition, we also obtain a combinatorial closed form for the Ehrhart series of $\\mathcal{M}_{pq}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09508",
  "author_total": 2,
  "authors": [
    {
      "name": "William Q. Erickson"
    },
    {
      "name": "Nicholas B. Jones"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/85vjk07v6zf2r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09508",
  "fulltext_url": null,
  "id": "85vjk07v6zf2r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09508",
  "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.AC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212111528",
  "pdf_url": "https://arxiv.org/pdf/2609.09508",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/85vjk07v6zf2r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09508",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09508"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A lattice path model for the volume of the Monge polytope",
  "title_text": "A lattice path model for the volume of the Monge polytope",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/85vjk07v6zf2r",
  "venue": "arXiv",
  "versions": []
}