{
  "abstract": "Let $n$ be a positive integer. A translation-invariant total order (TITO) with period $n$ is a total order of the integers that is invariant under translations by multiples of $n$. These structures arise naturally in the study of Coxeter groups. In particular, real $n$-TITOs are in bijection with biclosed sets of positive roots of the affine symmetric group $\\widetilde S_n$. Barkley and Defant recently introduced pattern avoidance for TITOs and used it to define the affine Tamari lattice. The enumeration of TITOs avoiding a single pattern of length $3$ is due to Crites and Barkley--Defant. We extend this work to TITOs avoiding two patterns. Our main results include a complete enumeration of TITOs that avoid a pair of patterns in $S_3\\times S_3$, as well as of TITOs that avoid a pair $(p, q)$ with $p \\in S_3 \\setminus \\{123, 321\\}$ and $q \\in S_4$. Furthermore, we provide an explicit construction of the inverse of the bijection between $312$-avoiding TITOs and noncrossing arc diagrams, thereby extending the combinatorial framework introduced by Barkley.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25656",
  "author_total": 1,
  "authors": [
    {
      "name": "Stella Jiahui Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vhbzx076xvhy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25656",
  "fulltext_url": null,
  "id": "2vhbzx076xvhy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25656",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214075371",
  "pdf_url": "https://arxiv.org/pdf/2609.25656",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vhbzx076xvhy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25656",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25656"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Enumerating pattern-avoiding translation-invariant total orders",
  "title_text": "Enumerating pattern-avoiding translation-invariant total orders",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/2vhbzx076xvhy",
  "venue": "arXiv",
  "versions": []
}