{
  "abstract": "For a finite sequence of positive integers $\\boldsymbol{a}=(a_1,\\dots,a_n)$, the restricted partition function $q_{\\boldsymbol{a}}(k)$ denote the number of nonnegative integer solutions to the equation $a_1x_1+a_2x_2+\\cdots +a_nx_n=k$. It is proved to be a quasi-polynomial of degree $n-1$. Write $q_{\\boldsymbol{a}}(k)=\\sum_{j=0}^{n-1}c_j(k)k^j$ with periodic coefficient functions $c_j$, and set $b_m=\\#\\{i:m\\mid a_i\\}$. In 2008, Beck, Sam, and Woods conjectured that the minimum period of $c_j(k)$ is $\\mathrm{lcm}\\{m:b_m>j\\}$. In this paper, we derive an exact root-of-unity formula for every coefficient function $c_j(k)$. The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of $c_j(k)$. Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02085",
  "author_total": 4,
  "authors": [
    {
      "name": "Feihu Liu"
    },
    {
      "name": "Jinlong Tang"
    },
    {
      "name": "Guoce Xin"
    },
    {
      "name": "Chen Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/et4hq490ey0xj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02085",
  "fulltext_url": null,
  "id": "et4hq490ey0xj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02085",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7172394542",
  "pdf_url": "https://arxiv.org/pdf/2608.02085",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/et4hq490ey0xj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2608.02085",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02085"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions",
  "title_text": "Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/et4hq490ey0xj",
  "venue": "arXiv",
  "versions": []
}