{
  "abstract": "The odd and even symmetric Dellac numbers are the specializations $a_n=f_n(0)$ and $e_n=f_n(1)$ of the Randrianarivony--Zeng family $f_n(s)=2^{-n}D_n(s)$. Using a finite Stieltjes-fraction truncation, we establish the sharp congruence $v_3(a_n)=1$ for $n\\ge2$ and obtain explicit congruences modulo $81$ for both sequences. We also establish universal eventual periodicity modulo arbitrary positive integers. For every integer $s\\ge0$ and $k\\ge1$, we give and analyse formulas for the \\emph{least} eventual period of $f_n(s)$ modulo $3^k$ and its \\emph{least starting index}. The period is governed by five linear factors in $s$; the onset follows a nine-class formula modulo $9$. In particular, for $k\\ge3$ the even symmetric Dellac period is three times the odd period and begins one index earlier. The proofs combine a finite Stieltjes-fraction certificate, a graded $3$-adic pole expansion, and an explicit finite top-degree certificate over $\\F_3$. Finite integer-arithmetic checks supplement the proofs. We also record the connection with the type-$B$ homogenized Linial arrangement.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05962",
  "author_total": 2,
  "authors": [
    {
      "name": "Tony Newton"
    },
    {
      "name": "Jiang Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ssdaf3gq1c5q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05962",
  "fulltext_url": null,
  "id": "7ssdaf3gq1c5q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05962",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05962",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ssdaf3gq1c5q.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05962",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05962"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact $3$-Adic Periods of Symmetric Dellac Numbers and Their Polynomial Extensions",
  "title_text": "Exact $3$-Adic Periods of Symmetric Dellac Numbers and Their Polynomial Extensions",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/7ssdaf3gq1c5q",
  "venue": "arXiv",
  "versions": []
}