{
  "abstract": "We introduce a new permutation statistic, called the \\emph{cross work}, on four important subsemigroups of the partial transformation semigroup $\\mathcal{PT}_n$: the symmetric group $\\mathcal{S}_n$, the semigroup of singular transformations $\\mathrm{Sing}_n$, the symmetric inverse semigroup $\\mathcal{I}_n$, and the semigroup $\\mathcal{POI}_n$ of partial order-preserving injective transformations. The cross work is defined via local distance interactions between adjacent positions and their images. Fundamental invariance properties are established, including inversion invariance under a natural symmetry condition and a characterization of conjugations preserving the statistic. Using elementary combinatorial methods, we derive closed formulas for the total and average cross work on $\\mathcal{S}_n$. We also determine the exact minimum and maximum values of the cross work, showing that the maximal value grows quadratically with $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34902",
  "author_total": 3,
  "authors": [
    {
      "name": "Nsibiet E. Udo"
    },
    {
      "name": "Victor A. Owai"
    },
    {
      "name": "Moses O. Francis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1ztj51cabpves.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34902",
  "fulltext_url": null,
  "id": "1ztj51cabpves",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34902",
  "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"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223989581",
  "pdf_url": "https://arxiv.org/pdf/2610.34902",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1ztj51cabpves.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34902",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34902"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cross work induced by transformations on the symmetric group and related transformation semigroups",
  "title_text": "Cross work induced by transformations on the symmetric group and related transformation semigroups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/1ztj51cabpves",
  "venue": "arXiv",
  "versions": []
}