{
  "abstract": "For $n\\ge2$, a good slope of a permutation $f$ of $\\mathbb{Z}_n=\\mathbb{Z}/n\\mathbb{Z}$ is an element $m\\in\\mathbb{Z}_n$ for which $x\\mapsto f(x)+mx$ is a permutation. We prove the sharp upper bound $φ(n)$ on the number of good slopes, where $φ$ is Euler's totient function. We classify and count the equality cases for prime powers and products of two distinct odd primes. Non-affine equality cases exist for every composite $n\\ge6$. We also prove the sharp bound $p_0-1$ for the size of a set of good slopes with invertible differences between distinct elements, where $p_0$ is the least prime divisor of $n$. Finally, we establish gap results below $φ(n)$ and apply our bounds to a construction of quantum Latin squares due to Ball and Simoens.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01465",
  "author_total": 1,
  "authors": [
    {
      "name": "Dipak K. Bhunia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9kg8a57v503n3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01465",
  "fulltext_url": null,
  "id": "9kg8a57v503n3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01465",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01465",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9kg8a57v503n3.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01465",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01465"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Extremal permutations of $\\mathbb{Z}_n$ with many good slopes and applications to quantum Latin squares",
  "title_text": "Extremal permutations of ℤ_n with many good slopes and applications to quantum Latin squares",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/9kg8a57v503n3",
  "venue": "arXiv",
  "versions": []
}