{
  "abstract": "Missing integers in partitions have been studied by many authors. A missing integer of a partition is a positive integer smaller than its largest part that does not occur as a part. We investigate parity biases for the number of missing integers in partitions and overpartitions. For every integer $c\\ge2$, we prove that the coefficients $a_c(n)$ of $(-cq;q)_\\infty/(-q;q)_\\infty$ are eventually strictly positive and provide an effectively computable threshold. The proof uses the circle method, combining a saddle-point analysis on the major arc with explicit uniform estimates on the minor arcs. Specializing to $c=2$ and $c=3$, and verifying the remaining finite cases, we show that partitions and overpartitions of $n$ with an even number of missing integers outnumber those with an odd number for $n>34$ and $n>27$, respectively. We further prove that, for $n>10$, the difference between the numbers of ordinary partitions with an even and an odd number of missing integers is smaller than the number of partitions of $n$ into distinct parts. These results confirm three conjectures of Bhoria, Eyyunni and Santra. We also give a combinatorial interpretation of $a_c(n)$ for every integer $c\\ge4$ in terms of colored partitions, generalizing the partition and overpartition interpretations of Bhoria, Eyyunni and Santra for $c=2$ and $c=3$, respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07908",
  "author_total": 2,
  "authors": [
    {
      "name": "Julia Q. D. Du"
    },
    {
      "name": "Gurinder Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/49ph9jmdks4gz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07908",
  "fulltext_url": null,
  "id": "49ph9jmdks4gz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07908",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07908",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/49ph9jmdks4gz.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07908",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07908"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The parity of the number of missing integers in partitions",
  "title_text": "The parity of the number of missing integers in partitions",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/49ph9jmdks4gz",
  "venue": "arXiv",
  "versions": []
}