{
  "abstract": "Kahane (arXiv:2607.11225) proves that for a fence poset $P$ of size $n$, $n!\\,Ω(P;t)=\\sum_{σ\\in S_n} t^{\\mathrm{bl}(σ)}$, with $\\mathrm{bl}(σ)$ a canonical block count, and conjectures (Conjecture 4.4) that the identity extends to the cell poset of any skew shape, with $\\mathrm{bl}$ replaced by a constructively-defined extension $\\widetilde{\\mathrm{bl}}$. We prove the conjecture for every $2\\times n$ rectangle -- the first genuinely two-dimensional case, where blocks are no longer path-intervals and the statistic is a correlated function of both rows. We introduce a two-parameter growth-state generating function $W_n(α,β;t)$ refining $\\sum_σt^{\\widetilde{\\mathrm{bl}}(σ)}$ by the rank $α$ of the bottom-row minimum and the rank $β$ of the rightmost top cell's block-root, and prove the closed form $W_n(α,β;t)=(α-1)!\\,B(n,β-2)\\,S_n(α,β;t)$ ($n\\ge 1$), where $B(n,k)$ is the Catalan triangle and $S_n$ a rising-factorial skeleton. The proof is an induction on $n$ through Kahane's column-insertion transition, whose load-bearing structural fact -- the block-root below a newly inserted top cell is always the global bottom-row minimum -- reduces the induction to a Catalan-triangle computation. Summing over $(α,β)$ telescopes to $C_n\\,t^{\\overline{n}}(t+1)^{\\overline{n}}$, which by MacMahon's product formula equals $(2n)!\\,Ω(R_{2,n};t)$. Every load-bearing step is a complete general-$n$ argument; the transition table and the inductive step are cross-checked by exhaustive enumeration (through $n=8$ and $n=9$) via an independent from-scratch implementation. General skew shapes and $m>2$ rows remain open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17561",
  "author_total": 1,
  "authors": [
    {
      "name": "Patrick White"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ahyv4shckxx7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17561",
  "fulltext_url": null,
  "id": "3ahyv4shckxx7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17561",
  "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"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213525555",
  "pdf_url": "https://arxiv.org/pdf/2609.17561",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3ahyv4shckxx7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17561",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17561"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A closed form for the block-statistic generating function of the $2\\times n$ rectangle: the two-row case of Kahane's conjecture",
  "title_text": "A closed form for the block-statistic generating function of the 2× n rectangle: the two-row case of Kahane's conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/3ahyv4shckxx7",
  "venue": "arXiv",
  "versions": []
}