{
  "abstract": "Let $A_k(n)$ and $B_k(n)$ be the numbers of ordered $k$-tuples of partitions with designated summands, respectively with designated summands and all parts odd, of total size $n$. Using an exact description of the generating function of designated partitions after a $3$-dissection, we classify for every $e$ the values of $k$ for which $A_k(3n+2)$ is divisible by $3^e$: this happens exactly when $k\\equiv0$ or $-5\\pmod{3^e}$ if $e\\le4$, and when $3^e\\mid k$ if $e\\ge5$. We obtain the corresponding exact $3$-adic and several $2$-adic results for $B_k$ and for $A_k$ along the progressions $4n+3$, a recursion for the classes $3^b(3n+s)$, and for $k=2$ the congruences $A_2(12n+10)\\equiv 27f_1^4f_6^3/f_2\\pmod{81}$ in generating-function form. We relate the numbers $A_k(n)$ modulo $3$ and $9$ to overpartitions, sums of squares and triangular numbers, give group-action proofs of some of the congruences, and close with several conjectures supported by computation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05623",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniel Yaqubi"
    },
    {
      "name": "Madjid Mirzavziri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3dtagxbm2taj3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05623",
  "fulltext_url": null,
  "id": "3dtagxbm2taj3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05623",
  "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/2611.05623",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3dtagxbm2taj3.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05623",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05623"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Partitions with designated summands: $k$-tuples and their congruences",
  "title_text": "Partitions with designated summands: k-tuples and their congruences",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:29Z",
  "url": "https://research.draconitic.com/paper/3dtagxbm2taj3",
  "venue": "arXiv",
  "versions": []
}