{
  "abstract": "The Clarke-Cormack-Burkowski lattice $\\mathcal{E}_n$ is the lattice of antichains of intervals of the chain $\\{0,1,\\dots,n-1\\}$, the algebraic structure underlying minimal-interval semantics, a query semantics used in information retrieval. We give an explicit, elementary anti-isomorphism between $\\mathcal{E}_n$ minus its top element and the Stanley lattice of Dyck words of semilength $n+1$ (an anti-isomorphism whose existence is implicit in work of Ferrari and Munarini), under which the rank of an antichain is $\\binom{n+1}{2}$ minus the area of the associated Dyck word. Hence the rank generating function of $\\mathcal{E}_n\\setminus\\{\\top\\}$ is the reverse of the Carlitz-Riordan $q$-Catalan polynomial. Using a local limit theorem for the area of random-walk excursions, we confirm unconditionally an asymptotic formula of Takács: the size $λ_n$ of the largest level set of $\\mathcal{E}_n$ satisfies $λ_n\\simκ\\,4^{n+1}/(n+1)^3$, where $κ\\approx1.0807$. Finally, for the antichains of intervals of an arbitrary finite poset $P$ we study the sweep dimension, the minimum number of linear orders needed to separate every interval from every element outside it. It equals the Dushnik-Miller dimension of the lattice of convex subsets of $P$, it is the least $k$ for which $P$ maps into $\\mathbb{R}^k$ so that every interval is the preimage of an upper orthant, and it lies between $\\max(2,w)$ and $2w$, where $w$ is the width of $P$, both bounds being tight. Antichains of intervals embed, preserving joins, into $k$-dimensional staircase regions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41385",
  "author_total": 2,
  "authors": [
    {
      "name": "Paolo Boldi"
    },
    {
      "name": "Sebastiano Vigna"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/487rksw09zw82.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41385",
  "fulltext_url": null,
  "id": "487rksw09zw82",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41385",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41385",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/487rksw09zw82.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41385",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41385"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The lattice of antichains of intervals: Dyck paths, level sets, and sweep dimension",
  "title_text": "The lattice of antichains of intervals: Dyck paths, level sets, and sweep dimension",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/487rksw09zw82",
  "venue": "arXiv",
  "versions": []
}