{
  "abstract": "Let $a_t(n)$ denote the coefficients of the Nekrasov--Okounkov hook length generating function $F_{t^2}(x)$, and let $b_t(n)$ denote the coefficients of the $t$-core partition generating function $C_t(x)$. We prove a sharp prime-power dichotomy: for every integer $t\\ge 2$, the congruence $$ a_t(n)\\equiv b_t(n)\\pmod t $$ holds for all $n\\ge 0$ if and only if $t$ is a prime power. When $t=p$ is prime, the congruence strengthens to modulo $p^2$. For $t=6$, and more generally for every integer that is not a prime power, the congruence fails. This identifies prime powers as the exact moduli for which these two partition-theoretic coefficient sequences are arithmetically equivalent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01422",
  "author_total": 1,
  "authors": [
    {
      "name": "Meenu Sharma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8k3w78estyj97.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01422",
  "fulltext_url": null,
  "id": "8k3w78estyj97",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01422",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7215736981",
  "pdf_url": "https://arxiv.org/pdf/2610.01422",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8k3w78estyj97.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01422",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01422"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Prime-Power Dichotomy for Nekrasov--Okounkov Hook Lengths and $t$-Core Partitions",
  "title_text": "A Prime-Power Dichotomy for Nekrasov--Okounkov Hook Lengths and t-Core Partitions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/8k3w78estyj97",
  "venue": "arXiv",
  "versions": []
}