{
  "abstract": "We study an operator $Θ$ on finite integer sequences, where $Θ(σ)_i$ counts the entries to the left of $σ_i$ that are strictly smaller than $σ_i$. This operator is a variant of the so-called Lehmer code. For every sequence $σ$, the image $Θ(σ)$ is an inversion sequence, and the restriction of $Θ$ to permutations of $[0,n-1]$ is a bijection onto inversion sequences of length $n$. We characterize the fixed points of $Θ$ by avoidance of the pattern $101$ together with a saturation condition, prove that they are counted by the Catalan numbers, and give an explicit recursive bijection with Dyck paths. We also show that the sequences whose first $Θ$-image is fixed are precisely those avoiding both $101$ and $201$. Finally, we prove finite stabilization for all inversion sequences, exhibit a family attaining the maximal stabilization time, and show that the second stabilization level is not closed under classical patterns.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24476",
  "author_total": 3,
  "authors": [
    {
      "name": "Julian Allagan"
    },
    {
      "name": "Shanzhen Gao"
    },
    {
      "name": "Benjamin Testart"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8w3ncrk4ng6pe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.24476",
  "fulltext_url": null,
  "id": "8w3ncrk4ng6pe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24476",
  "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": "W7204235360",
  "pdf_url": "https://arxiv.org/pdf/2608.24476",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8w3ncrk4ng6pe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2608.24476",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24476"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization",
  "title_text": "Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/8w3ncrk4ng6pe",
  "venue": "arXiv",
  "versions": []
}