{
  "abstract": "A permutation $σ\\in\\mathbb{S}_n$ of objects $\\{1,\\ldots,n\\}$ is called \\emph{late-growing} if its partial sums never exceed the respective partial sums of the `average' permutation (with the arithmetic mean $n^{-1}\\cdot\\sum_{i=1}^n i = (n+1)/2$ of $1,\\ldots,n$ on average at each entry $σ(i)$): $\\sum_{i=1}^k σ(i) \\leqslant k\\cdot (n+1)/2$ for all positive $k\\leqslant n$. We prove that for $n=2p-1$ odd, the set $Φ_p\\subseteq \\mathbb{S}_{2p-1}$ of late-growing permutations is partitioned into parity-even vs odd elements, $Φ_p = Φ_p^{+} \\sqcup Φ_p^{-}$, in an almost symmetric way: $| Φ_p^{+} | - | Φ_p^{-} | = (-1)^{p+1}$. The proof is constructive: by viewing the permutations $σ\\inΦ_p$ as constrained walks on the integral lattice $\\mathbb{Z}$, we describe two filters that pair all but one element in $Φ_p^{+}$ versus $Φ_p^{-}$; the remaining walk $σ^*$ with monotonically growing step amplitudes determines the dominant parity subset.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35779",
  "author_total": 2,
  "authors": [
    {
      "name": "Kian C. Shah"
    },
    {
      "name": "Arthemy V. Kiselev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d74335dk1adqr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35779",
  "fulltext_url": null,
  "id": "d74335dk1adqr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35779",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223928608",
  "pdf_url": "https://arxiv.org/pdf/2610.35779",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d74335dk1adqr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.35779",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35779"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "The counts of parity-odd vs even late-growing permutations in the group $\\mathbb{S}_{2p-1}$ differ by $(-1)^p$",
  "title_text": "The counts of parity-odd vs even late-growing permutations in the group 𝕊_2p−1 differ by (−1)^p",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/d74335dk1adqr",
  "venue": "arXiv",
  "versions": []
}