{
  "abstract": "In 2020, Brualdi and Dahl introduced the discrete derivative $σ'(i)=σ(i+1)-σ(i)$ of a permutation $σ\\in\\Sym_n$. In this paper, we focus on the second derivative of a permutation, $σ''(i)=σ(i+1)+σ(i-1)-2σ(i)$, and the inflections it determines. We show that for even $n$, the only distinct permutations with the same second derivative are the identity and its reverse, while for odd $n$, the second derivative is not injective even after excluding them. We then study which sets of pairs of integers $(i,j)$ are the inflection set of some permutation and whether a given set is witnessed in $\\Sym_n$ for infinitely many $n$. We show that exactly $14$ permutations of $[n]$ have no inflections when $n\\ge5$. For a single inflection $\\{(k,m)\\}$, we determine all $n$ for which it is witnessed in $\\Sym_n$: we prove it is always witnessed, and it is witnessed for only finitely many $n$ if and only if $m\\ge 2k+1$. In that case, all such $n$ satisfy $2m-2k+1\\le n\\le 2m-k+2$ (apart from $k=2$, $n=m+1$), and, perhaps surprisingly, this range has gaps once $k\\ge 10$, the smallest occurring at $(k,m,n)=(10,21,30)$. Finally, we characterize maximally inflected permutations, whose second derivative alternates in sign, and show that their number $|\\mathcal M_n|$ satisfies $2E_n\\le|\\mathcal M_n|\\le(2/3)^{\\lfloor n/3\\rfloor-1}\\,n!$, where $E_n$ is the Euler zigzag number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38237",
  "author_total": 2,
  "authors": [
    {
      "name": "Gordon Rojas Kirby"
    },
    {
      "name": "Mariana Smit Vega Garcia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8zn64ntrgfh8d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38237",
  "fulltext_url": null,
  "id": "8zn64ntrgfh8d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38237",
  "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/2610.38237",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8zn64ntrgfh8d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38237",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38237"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Inflection sets of permutations",
  "title_text": "Inflection sets of permutations",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/8zn64ntrgfh8d",
  "venue": "arXiv",
  "versions": []
}