{
  "abstract": "Sorting By Transpositions (SBT) seeks the minimum number of transpositions required to sort a permutation $π$ on $n$ symbols into the identity $ι$. Let $N=n+1$. A cyclic-target pair $(ω,β)$ consists of an even permutation $ω$ and an $N$-cycle $β$ for which $ρ=ωβ$ is an $N$-cycle. An SBT instance is the special case $(\\barι{\\barπ}^{-1},\\barπ)$, where $\\barπ$ and $\\barι$ encode $π$ and $ι$, and $\\barι{\\barπ}^{-1}\\barπ=\\barι$. For a prescribed fixed-point-free cycle type, fixed-content words encode $ω$, with colors distinguishing cycles and ranks recording their orientations relative to $β$. A word is realizable exactly when $ρ=ωβ$ is an $N$-cycle. Permutations of equal-part colors and shifts of rank origins form auxiliary symmetries that, together with word rotation and position reflection coupled to rank inversion, define a twisted dihedral action. Its orbits are twisted bracelets, and its realizable orbits correspond bijectively to extended-toric equivalence classes of cyclic-target pairs, where reflection is adjoined to classical toric equivalence. The correspondence yields a Burnside identity, and our method generates one encoding word for each such class. The transposition diameter $TD(n)$ is the largest transposition distance in $S_n$. Combining fixed-point contraction and reductions of the ambient instance space based on cycle structure with exhaustive verification of every remaining twisted bracelet, we prove $TD(16)=9$, closing a twenty-five-year gap. This result also yields $TD(19)=11$ and, for every $n\\equiv1\\pmod{3}$ with $n\\geq16$, $TD(n)\\leq\\left\\lfloor(2n-2)/3\\right\\rfloor-1$, improving the previous general upper bound by one for these $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17493",
  "author_total": 4,
  "authors": [
    {
      "name": "Luiz A. G. Silva"
    },
    {
      "name": "Luis A. B. Kowada"
    },
    {
      "name": "Noraí R. Rocco"
    },
    {
      "name": "Maria E. M. T. Walter"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d7dswsknjrmvt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17493",
  "fulltext_url": null,
  "id": "d7dswsknjrmvt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17493",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213417731",
  "pdf_url": "https://arxiv.org/pdf/2609.17493",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d7dswsknjrmvt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.17493",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17493"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Twisted Bracelets for Sorting by Transpositions: the Transposition Diameter of $S_{16}$",
  "title_text": "Twisted Bracelets for Sorting by Transpositions: the Transposition Diameter of S_16",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/d7dswsknjrmvt",
  "venue": "arXiv",
  "versions": []
}