{
  "abstract": "Andrews and Newman \\cite{andrewsnewmanI} introduced and studied ``mex,'' function for an integer partition $λ$ of a positive integer $n$. The related function $σ^{\\mathrm{mex}}(n)$ was subsequently defined, and its generating function and other properties have been studied by several authors. In this article, for a positive integer $\\ell \\geq 2$, we define the minimal excludant of an $\\ell$-regular partition $λ$ of a positive integer $n$. We then define $σ_\\ell^{\\mathrm{mex}}(n)$, the sum of the minimal excludants over all $\\ell$-regular partitions of $n$, and obtain explicit generating functions for $σ_\\ell^{\\mathrm{mex}}(n)$ in the cases $\\ell = 2$ and $\\ell = 3$, expressed as eta-quotients. We prove that the sequence $σ_\\ell^{\\mathrm{mex}}(n)$ is weakly increasing for every $\\ell$, and we obtain Hardy--Ramanujan type asymptotic formulas for $σ_2^{\\mathrm{mex}}(n)$ and $σ_3^{\\mathrm{mex}}(n)$ as $n \\to \\infty$. Additionally, we completely determine the parity of $σ_2^{\\mathrm{mex}}(n)$, showing that it is odd precisely at generalized pentagonal numbers, and we use this characterization to compute the density of $n$ for which $σ_2^{\\mathrm{mex}}(n)$ is odd versus even. Finally, for $σ_3^{\\mathrm{mex}}(n)$, we prove that it is odd infinitely often, and we establish a quantitative lower bound, showing that the number of such $n$ up to $X$ grows at least like $\\log \\log X$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06072",
  "author_total": 1,
  "authors": [
    {
      "name": "Nabin Kumar Meher"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9cg5gt1berjej.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06072",
  "fulltext_url": null,
  "id": "9cg5gt1berjej",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06072",
  "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.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06072",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9cg5gt1berjej.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06072",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06072"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Minimal excludant of $\\ell$-regular partition",
  "title_text": "The Minimal excludant of ℓ-regular partition",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/9cg5gt1berjej",
  "venue": "arXiv",
  "versions": []
}