{
  "abstract": "We consider the family of operators of stack sorting (s_m)_{m >= 1} acting on words according to the following rule: s_m(w) is the output produced by the usual West algorithm of stack sorting on the input word w, but allowing for at most m repetitions of the same letter stacked in succession. Our first main theorem gives an explicit formula for the action of the operator s_m on binary words w = a^p b a^q (with distinct letters a > b): s_m(a^p b a^q) = a^max(p-m,0) b a^min(p,m)+q, d_m(a^k b) = ceil(k/m). From the above formulas it follows that for every m < m' we have d_m(w) >= d_{m'}(w) for binary words w = a^k b, and moreover, the ratio d_m(w)/d_{m'}(w) is always possible to be chosen exactly m'/m, so the ratios d_m/d_{m'} are unbounded for this family on the specified class -- this is an explicit form of the expected separation-speedup phenomenon. Additionally, we provide explicit constructions of words of length 4 for which s_m and s_{m'} do not commute, and words of length 7 for which the speed d_m is not monotonic in m. Further, we provide a full recursive characterization -- generalizing the classical 231-avoidance result of Knuth and West -- of words which can be sorted using s_m only once, and then we employ it to show that there cannot exist an m-independent classical pattern avoidance condition for one-pass sortable words. Finally, we give a complete proof of the inequality d_1(w) >= d_m(w) >= d_infinity(w) for all m for two-letter words, together with a structural result establishing it for a single pass in general; nevertheless, we show that the family of operators (s_m) does not, after all, lie entirely between s_1 and s_infinity, exhibiting for every n >= 7 an explicit word w_n with d_1(w_n) = n-4 < n-3 = d_m(w_n) for every m >= 2, including m = infinity, so that the inequality above is false in general.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05681",
  "author_total": 1,
  "authors": [
    {
      "name": "Fayzan Khan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/av8xf6yerfwss.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05681",
  "fulltext_url": null,
  "id": "av8xf6yerfwss",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05681",
  "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": "W7212042800",
  "pdf_url": "https://arxiv.org/pdf/2609.05681",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/av8xf6yerfwss.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05681",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05681"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Stack Sorting on Words with Bounded-Repetition",
  "title_text": "Stack Sorting on Words with Bounded-Repetition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/av8xf6yerfwss",
  "venue": "arXiv",
  "versions": []
}