{
  "abstract": "We extend the recent work of Caster--Guyer--Samper (2026+) to establish a combinatorial interpretation of the $\\textbf{cd}$-index for all $S$-partitionable posets, which includes all polytopes and $S$-shellable spheres. This interpretation is inductive on rank, but does not require passing to the order complex of the poset. We also provide topological interpretations of each $\\textbf{cd}$-monomial. Furthermore, we establish a recurrence on the $\\textbf{cd}$-indices of shelling components of the $d$-simplex that does not require the use of André permutations, a unimodality result for $\\textbf{cd}$-words ending in $\\textbf{d}$ for shelling components of the $d$-simplex and the $d$-cube, and provide two families of subdivisions of the boundary of the $d$-simplex whose sequence of $\\textbf{cd}$-indices is unbounded on all but the monomial $\\textbf{c}^{d+1}$ and all but the two monomials $\\textbf{c}^{d+1}$ and $\\textbf{d}\\textbf{c}^{d-1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25491",
  "author_total": 2,
  "authors": [
    {
      "name": "Dan Guyer"
    },
    {
      "name": "Alex Waugh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ew0cnscm1wg21.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25491",
  "fulltext_url": null,
  "id": "ew0cnscm1wg21",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25491",
  "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": "W7223059511",
  "pdf_url": "https://arxiv.org/pdf/2610.25491",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ew0cnscm1wg21.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25491",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25491"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Interpretations of the $\\textbf{cd}$-index",
  "title_text": "Interpretations of the cd-index",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/ew0cnscm1wg21",
  "venue": "arXiv",
  "versions": []
}