{
  "annotation_id": "8973d37c-3954-4213-b743-fb1c1e5e7875",
  "date_created": "2026-02-17T05:53:20.412000Z",
  "date_modified": "2026-02-17T05:53:20.412000Z",
  "file_hash": "8d41335daea7ade67ff0a685741c4fc48eb2e3a7a16d894c4d5f4599650e5b51",
  "private": false,
  "record": {
    "abstract": "Let $p(n)$ denote the partition function and define $p(n,k)=\\sum_{j=0}^{k}\\binom{n-j}{k-j}p(j)$ where $p(0)=1$. We prove that $p(n,k)$ is unimodal and satisfies $p(n,k) \u003c \\frac{2.825}{\\sqrt{n}}\\, 2^n $ for fixed $n\\ge 1$ and all $1\\le k\\le n$. This result has an interesting application: the minimal dimension of a faithful module for a $k$-step nilpotent Lie algebra of dimension $n$ is bounded by $p(n,k)$ and hence by $\\frac{3}{\\sqrt{n}}\\, 2^n $, independently of $k$. So far only the bound $n^{n-1}$ was known. We will also prove that $p(n,n-1)\u003c\\sqrt{n}\\exp(\\pi\\sqrt{2n/3})$ for $n\\ge 1$ and $p(n-1,n-1)\u003c\\exp (\\pi\\sqrt{2n/3} )$.",
    "arxiv_id": "2601.09472",
    "authors": [
      "Dietrich Burde"
    ],
    "categories": [
      "math.NT"
    ],
    "doi": "10.1007/s002290070002",
    "journal_ref": "Manuscripta mathematica, Vol. 103 (2000), Issue 4, 435-446",
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Estimates on binomial sums of partition functions",
    "url": "https://arxiv.org/abs/2601.09472",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "b6067172-226c-454a-8b1c-4ace4ddc1cdd",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}