{
  "abstract": "Let $F=F(X)$ be a free group of finite rank, with palindromic length taken with respect to the fixed basis $X$. We embed $F$ as the index-two subgroup of the universal Coxeter group $W=F\\rtimes_θ\\langle t\\mid t^2=1\\rangle$, where $θ(x)=x^{-1}$ for $x\\in X$, and prove $\\mathrm{pl}(g)=\\min{\\ell_T(g),\\ell_T(gt)}$. Dyer's deletion theorem then identifies reflection length with the minimum number of unmatched positions in a noncrossing equal-label partial matching on a reduced Coxeter word. This gives an $O(n^3)$-time, $O(n^2)$-space algorithm for palindromic length, together with recovery of an optimal palindromic factorization. The matching model also gives a structural characterization. For every ordered full binary tree with $k$ leaves we define a literal word template whose leaves are palindromes and whose internal vertices carry arbitrary words. A reduced word $w$ represents an element of palindromic length at most $k$ if and only if $w$ is a literal instance of one of these templates. Hence the $C_{k-1}$ ordered binary-tree shapes give a complete finite family for each fixed $k$. For $k=4$ the five templates are exactly the five forms proposed by Frid, proving the completeness of that list. A companion Lean 4 development verifies the four-palindrome classification end to end for every finite rank, including the ordinary reduced-word formulation and the literal five-form conclusion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17027",
  "author_total": 1,
  "authors": [
    {
      "name": "Junjie Liao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2mp5ww5bdrffy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17027",
  "fulltext_url": null,
  "id": "2mp5ww5bdrffy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17027",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7213428062",
  "pdf_url": "https://arxiv.org/pdf/2609.17027",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2mp5ww5bdrffy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17027",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17027"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Palindromic Length in Free Groups: Reflections, Noncrossing Matchings, and Catalan Forms",
  "title_text": "Palindromic Length in Free Groups: Reflections, Noncrossing Matchings, and Catalan Forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/2mp5ww5bdrffy",
  "venue": "arXiv",
  "versions": []
}