{
  "abstract": "Let $P_d$ be the convex hull of the permutations of $\\{1,\\ldots,d\\}$ that avoid the patterns $132$ and $213$, and let $H_d(t)$ be its Ehrhart $h^*$-polynomial. We prove that all zeros of $H_d(t)$ are real and negative for $3\\le d\\le1000$ and for every $d\\ge2^{62}$; for $d\\le2$ the polynomial is constant. Expanding $H_d(t)$ in refined Eulerian polynomials does not suffice, since a coordinate is already negative at $d=7$. Let $A_{d,i}(t)$ enumerate all permutations of $\\{1,\\ldots,d\\}$ with first entry $i$ by number of descents. For $c\\ge0$, we prove that every nonnegative linear combination of $A_{d,1},\\ldots,A_{d,d}$ and the adjacent differences $A_{d,r+1}-cA_{d,r}$, $2\\le r\\le d-2$, is real-rooted for all $d\\ge4$ simultaneously if and only if $c\\le2-\\sqrt3$. This reduces real-rootedness to the positivity of explicit weighted tails of the coordinates. An exact certificate checks these inequalities for $d\\le1000$, and coefficient estimates based on Darroch's theorem on the mode of a sum of Bernoulli variables prove them for $d\\ge2^{62}$; the range $1001\\le d<2^{62}$ remains open. Along the way we identify $P_d$, up to lattice equivalence, with twice a chain-poset permutahedron and with the Newton polytope of a discriminant, which transfers the known cube structure and volume of these models to $P_d$. We also show that the coefficient of $t^{d-1}$ in $H_d(t)$ counts strong tournament score sequences for $d\\ge3$, confirming a conjecture of Rehberg in this range.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06096",
  "author_total": 1,
  "authors": [
    {
      "name": "Pedro M. M. de Castro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4q08ks37bghad.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06096",
  "fulltext_url": null,
  "id": "4q08ks37bghad",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06096",
  "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.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212032212",
  "pdf_url": "https://arxiv.org/pdf/2609.06096",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4q08ks37bghad.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.06096",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06096"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the real-rootedness of Ehrhart $h^*$-polynomials of $(132,213)$-avoiding permutahedra",
  "title_text": "On the real-rootedness of Ehrhart h^*-polynomials of $(132,213)$-avoiding permutahedra",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/4q08ks37bghad",
  "venue": "arXiv",
  "versions": []
}