{
  "abstract": "Let $π=π_1\\cdots π_n$ be a permutation. We say that $π$ is $run$-$sorted$ if $π_1=1$ and the entries immediately following the descent positions of $π$ form an increasing sequence. Let $\\mathcal{R}_n$ denote the set of run-sorted permutations of length $n$, which has cardinality given by the Bell number $B_{n-1}$ for all $n \\geq 1$. In this paper, we consider the joint distribution $A_n(q,u)$ on $\\mathcal{R}_n$ for the parameters tracking the numbers of inversions and runs leading to a new polynomial generalization of the Bell numbers. Among our results, we find a general recurrence for $A_n(q,u)$, from which one may derive explicit formulas for the total numbers of inversions or runs in all the members of $\\mathcal{R}_n$ as well as for the sign-balance on $\\mathcal{R}_n$ of either parameter. A simple expression for the Eulerian generating function for $A_n(q,1)$ may be found upon making use of Gessel's $q$-exponential formula which can be extended to general $u$. Finally, a formula is found by a direct argument for the maximum number of inversions within a member of $\\mathcal{R}_n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28674",
  "author_total": 3,
  "authors": [
    {
      "name": "Toufik Mansour"
    },
    {
      "name": "Olivia Nabawanda"
    },
    {
      "name": "Mark Shattuck"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1jr5s4pq1zqb1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28674",
  "fulltext_url": null,
  "id": "1jr5s4pq1zqb1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28674",
  "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": "W7214290564",
  "pdf_url": "https://arxiv.org/pdf/2609.28674",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1jr5s4pq1zqb1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28674",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28674"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Distribution of the inversion statistic on run-sorted permutations",
  "title_text": "Distribution of the inversion statistic on run-sorted permutations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/1jr5s4pq1zqb1",
  "venue": "arXiv",
  "versions": []
}