{
  "abstract": "Unit interval graphs admit a classical Catalan encoding by area sequences, or equivalently by Dyck paths. We introduce a new statistic on these graphs, called peak-nesting, defined as the largest number of peak-cliques containing a common vertex. Through the correspondence with area sequences, peak-nesting also defines a new statistic on Dyck paths. We construct a bijection between Dyck paths and bicolored Motzkin paths which simultaneously records peak-nesting and the number of peaks. This yields a refinement of Touchard's identity, coefficient formulas involving Dyck paths of given height, and new combinatorial interpretations for several sequences recorded in the OEIS. We study the peak and peak-nesting polynomials over all unit interval graphs and over the subclasses of connected, Abelian, $2$-nested, and reduced unit interval graphs. For these families we obtain closed formulas, recurrences, and, for the symmetric cases, nonnegative expansions in the gamma-basis with explicit combinatorial interpretations. We also investigate questions regarding the location of zeros and the log-concavity of the corresponding coefficient sequences: some of the polynomial families form generalized Sturm sequences, whereas others fail to be real-rooted but appear nevertheless to have log-concave coefficients. Finally, we relate the peak-clique presentation of a unit interval graph to lattice path matroids.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38084",
  "author_total": 2,
  "authors": [
    {
      "name": "Per Alexandersson"
    },
    {
      "name": "Leonardo Saud Maia Leite"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0yvsqdgjm7b91.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38084",
  "fulltext_url": null,
  "id": "0yvsqdgjm7b91",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38084",
  "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"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214921255",
  "pdf_url": "https://arxiv.org/pdf/2609.38084",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0yvsqdgjm7b91.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:36Z",
      "native_id": "2609.38084",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38084"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Peaks and peak-nestings on unit interval graphs",
  "title_text": "Peaks and peak-nestings on unit interval graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:36Z",
  "url": "https://research.draconitic.com/paper/0yvsqdgjm7b91",
  "venue": "arXiv",
  "versions": []
}