{
  "abstract": "The minimal excludant (mex) of an integer partition $λ$ is the smallest positive integer that is not a part of $λ$. In this paper we study an analogue of this statistic for $\\ell$-regular partitions with $\\ell\\geq2$, that is, partitions in which no part is divisible by $\\ell$. Recently, Meher \\cite{Meher} defined the minimal excludant for $\\ell$-regular partitions. Motivated by Meher \\cite{Meher}, for an $\\ell$-regular partition we define its least $r$-gap as the smallest positive integer not divisible by $\\ell$ that appears fewer than $r$ times as a part, and we obtain the two-variable generating function for the number of $\\ell$-regular partitions of $n$ with a given least $r$-gap. Let $σ^{\\mex}_{\\ell}(n)$, and $σ^{\\mex}_{\\ell,r}(n)$ be the sum of the of the mex values over all $\\ell$-regular partitions of $n$ and least $r$-gaps over all $\\ell$-regular partitions of $n$ respectively. Using Jacobi's triple product identity, we derive explicit expressions for the generating function of $σ^{\\mex}_{\\ell}(n)$, and $σ^{\\mex}_{\\ell,r}(n)$, with separate formulas for even and odd $\\ell$. We also show that the sequence $\\{σ^{\\mex}_{\\ell,r}(n)\\}_{n\\ge0}$ is weakly increasing for every $\\ell\\ge2$ and $r\\ge1$. Also, using Ingham's Tauberian theorem, we establish Hardy--Ramanujan type asymptotic formulas for $σ^{\\mex}_{\\ell,r}(n)$ when $\\ell=4,5,7$ and $r\\ge1$ is arbitrary. Lastly, we prove $σ^{\\mex}_{\\ell,r}$ 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": "2611.07871",
  "author_total": 3,
  "authors": [
    {
      "name": "Tejash Priya"
    },
    {
      "name": "Sourav Bhowmick"
    },
    {
      "name": "Nabin Kumar Meher"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d2e033bb54c2n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07871",
  "fulltext_url": null,
  "id": "d2e033bb54c2n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07871",
  "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/2611.07871",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d2e033bb54c2n.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07871",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07871"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The minimal excludant and the least $r$-gaps of $\\ell$-regular partitions",
  "title_text": "The minimal excludant and the least r-gaps of ℓ-regular partitions",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/d2e033bb54c2n",
  "venue": "arXiv",
  "versions": []
}