{
  "abstract": "The Delannoy numbers $d(n,k)$ count lattice paths from $(0,0)$ to $(n-k,k)$ using steps $(1,0),(0,1)$ and $(1,1)$. This paper introduces a graded poset $\\mathcal{D}_n$ on the Delannoy paths ending on the line $x+y=n$, whose rank-generating function is the Delannoy polynomial $d_n(x)=\\sum_{k=0}^n d(n,k)x^k$. We prove that $\\\\mathcal{D}_n$ is a self-dual lattice, which is call the Delannoy lattice. By establishing an explicit symmetric Boolean decomposition, we show that $\\mathcal{D}_n$ is a symmetric Boolean order, thereby recovering the $γ$-positivity of $d_n(x)$. Such a decomposition is refined to a symmetric chain decomposition with the chain cover property, and is applied to determine all maximum antichains. We also investigate other combinatorial aspects of $\\mathcal{D}_n$, including supersolvability, the Möbius number, characteristic polynomials, and zeta polynomials.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30350",
  "author_total": 2,
  "authors": [
    {
      "name": "Xi Chen"
    },
    {
      "name": "Yuxian Dong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dbt1n2sefqsyq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30350",
  "fulltext_url": null,
  "id": "dbt1n2sefqsyq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30350",
  "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"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214632743",
  "pdf_url": "https://arxiv.org/pdf/2609.30350",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dbt1n2sefqsyq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30350",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30350"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Combinatorial aspects of the Delannoy Lattice",
  "title_text": "Combinatorial aspects of the Delannoy Lattice",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/dbt1n2sefqsyq",
  "venue": "arXiv",
  "versions": []
}