{
  "abstract": "The palindromic length $PL(u)$ of a nonempty finite word $u$ is the least number of nonempty palindromes whose concatenation is $u$. For an infinite word $w$, let $P_w(n)$ be the number of nonempty palindromic prefixes of $w[1,n]$, and let $T_w(n)$ consist of those prefixes $w[1,j]$, $1\\leq j\\leq n$, whose palindromic length equals the maximum attained among the nonempty prefixes of $w[1,j]$. We prove the finite inequality $|T_w(n)|\\geq P_w(n)$ for every $n\\geq1$. In particular, if $w$ has infinitely many palindromic prefixes, then \\[ \\liminf_{n\\to\\infty}\\frac{|T_w(n)|}{P_w(n)}\\geq1. \\] The coefficient $1$ is optimal, already for a nonconstant periodic word. The proof uses chains of occurrences connected by palindromic factors, together with trimming and reflection arguments that preserve lower bounds on palindromic length.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15146",
  "author_total": 1,
  "authors": [
    {
      "name": "Josef Rukavicka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8d9bs93f0vrbw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15146",
  "fulltext_url": null,
  "id": "8d9bs93f0vrbw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15146",
  "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",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7213363807",
  "pdf_url": "https://arxiv.org/pdf/2609.15146",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8d9bs93f0vrbw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15146",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15146"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A lower bound on the density of prefixes with maximal palindromic length",
  "title_text": "A lower bound on the density of prefixes with maximal palindromic length",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/8d9bs93f0vrbw",
  "venue": "arXiv",
  "versions": []
}