{
  "abstract": "The rows and columns of the character table of the symmetric group $S_n$ are both naturally indexed by partitions of $n$. Let $D(n)$ denote the number of conjugacy classes of $S_n$ whose column contains no zero entry. The identity column is always zero-free, so $D(n)\\geq 1$. It is known that $D(n)\\ll n^2$. We prove that $D(n)\\ll n^{3/4}$. Second, we prove for almost all positive integers $n$ that $D(n)\\ll_B n^{1/2}(\\log n)^B$ for every $B>5/6$, with a quantitative bound for the exceptional set, using work of Matomäki and Radziwill. Finally, we offer a heuristic supporting our conjecture that $D(n)\\ll_{\\varepsilon} n^{\\varepsilon}$. AxiomProver formalized the results in this paper in Lean assuming preexisting literature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27718",
  "author_total": 4,
  "authors": [
    {
      "name": "Colin Defant"
    },
    {
      "name": "Sidharth Hariharan"
    },
    {
      "name": "Kenny Lau"
    },
    {
      "name": "Ken Ono"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2cq2e9hmdc5rw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27718",
  "fulltext_url": null,
  "id": "2cq2e9hmdc5rw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27718",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7204816626",
  "pdf_url": "https://arxiv.org/pdf/2608.27718",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2cq2e9hmdc5rw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27718",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27718"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Zero-free columns in character tables of symmetric groups",
  "title_text": "Zero-free columns in character tables of symmetric groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/2cq2e9hmdc5rw",
  "venue": "arXiv",
  "versions": []
}