{
  "abstract": "If $\\boldsymbol{w} = (w_1,w_2,\\dots)$ is a sequence in $\\mathbb{N}=\\{1,2,\\dots\\}$, the partial Bell polynomials based on $\\boldsymbol{w}$ are $B_{n,k}$ for $k \\in \\mathbb{N}$ and $n\\in\\{k,k+1,\\dots\\}$. Let $F(z) = \\sum_{n=1}^{\\infty} (w_n/n!)z^n$ be the exponential generating function for $\\boldsymbol{w}$, and assume the radius of convegence is positive $R>0$. Then $F(z)^k = \\sum_{n=k}^{\\infty} (k!/n!) z^n B_{n,k}$ for $|z|<R$. Alternatively, defining $Q_{k,n} = (k!/n!)B_{n,k}$, we have $Q_{1,n} = w_n/n!$, and $Q_{k+1,n}=\\sum_{m=1}^{n-k} Q_{1,m} Q_{k,n-m}$ for $k\\geq 1$. Let us say that the Cramér-type large deviation property holds if $$ \\lim_{\\substack{n \\to \\infty\\\\ k/n \\to κ}} \\frac{1}{n}\\, \\ln\\left(Q_{k,n}\\right)\\, =\\, \\mathcal{G}(κ)\\, ,$$ for every $κ\\in (0,1)$, where $\\mathcal{G}(κ)=\\inf_{r \\in (0,R)} (κ\\ln(F(r))-\\ln(r))$. The (Hardy-Ramanujan) Erdös induction argument suggests this should generally be true as long as two technical conditions are true: one an initial step, and the other a condition for small densities $κ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06281",
  "author_total": 2,
  "authors": [
    {
      "name": "Sophia Li"
    },
    {
      "name": "Shannon Starr"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0s1ccvzz7r5v6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06281",
  "fulltext_url": null,
  "id": "0s1ccvzz7r5v6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06281",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7212043198",
  "pdf_url": "https://arxiv.org/pdf/2609.06281",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0s1ccvzz7r5v6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.06281",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06281"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "About the Cramér Large Deviation Property for Bell Polynomials",
  "title_text": "About the Cramér Large Deviation Property for Bell Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/0s1ccvzz7r5v6",
  "venue": "arXiv",
  "versions": []
}