{
  "abstract": "Given an $n\\times n$ matrix $A=(a_{i,j})$ and subsets $I,J$ of $[1,n]:=\\{1,\\dotsc,n\\}$, let $A_{I,J}=(a_{i,j})_{i\\in I,j\\in J}$ denote the $(I,J)$-submatrix of $A$. We prove that, for every totally nonnegative matrix $A$ and every $h=1,\\dotsc,n-1$, \\begin{equation} \\operatorname{perm}(A_{[1,h],[1,h]}) \\operatorname{perm}(A_{[h+1,n],[h+1,n]}) \\;\\geq\\; \\operatorname{perm}(A_{[1,n]\\cap 2\\mathbb Z,[1,n]\\cap 2\\mathbb Z}) \\operatorname{perm}(A_{[1,n]\\smallsetminus 2\\mathbb Z,[1,n]\\smallsetminus 2\\mathbb Z}). \\label{eq:absmain} \\end{equation} In particular, these inequalities hold for the $0$-$1$ totally nonnegative matrices arising as antiadjacency matrices of unit interval orders. Our proof is based on explicit Bruhat-order comparisons between two distinguished families of permutations. Let $\\mathfrak{A}_n$ be the Young subgroup $\\mathfrak{S}_{1,\\ldots,\\left\\lfloor\\frac{n}{2}\\right\\rfloor} \\times \\mathfrak{S}_{\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1,\\ldots,n}$ and let $\\mathfrak{B}_n$ be the set of parity-alternating permutations \\begin{equation*} \\{w=w_1\\cdots w_n\\in\\mathfrak{S}_n \\mid i \\text{ and } w_i \\text{ have the same parity}\\}. \\end{equation*} For the balanced case $h=\\left\\lfloor\\frac{n}{2}\\right\\rfloor$, we construct a bijection $f_n:\\mathfrak{A}_n\\rightarrow\\mathfrak{B}_n$ satisfying $w\\leq f_n(w)$ in Bruhat order. We then extend this construction to arbitrary $h$ by means of maps $g_{n,h}$ satisfying the corresponding Bruhat-order comparison. These maps yield the inequalities for all totally nonnegative matrices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04809",
  "author_total": 3,
  "authors": [
    {
      "name": "Sihong Pan"
    },
    {
      "name": "Mark Skandera"
    },
    {
      "name": "Jiayuan Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6asmsmnkkmsm2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04809",
  "fulltext_url": null,
  "id": "6asmsmnkkmsm2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04809",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.04809",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6asmsmnkkmsm2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.04809",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04809"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "From Unit Interval Orders to Totally Nonnegative Matrices: Permanental Inequalities and Bruhat-Increasing Bijections",
  "title_text": "From Unit Interval Orders to Totally Nonnegative Matrices: Permanental Inequalities and Bruhat-Increasing Bijections",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/6asmsmnkkmsm2",
  "venue": "arXiv",
  "versions": []
}