AI 中文总结
该论文证明了 Kahane 猜想对 $2\times n$ 矩形成立，通过引入双参数生成函数并利用列插入归纳得到闭式，最终验证了块统计生成函数的乘积公式。
AI 中文摘要
Kahane ( arXiv:2607.11225 ) 证明了对于大小为 $n$ 的栅栏偏序集 $P$，有 $n!\\,\Omega(P;t)=\sum_{\sigma\in S_n} t^{\mathrm{bl}(\sigma)}$，其中 $\mathrm{bl}(\sigma)$ 是典范块计数，并猜想（猜想 4.4）该恒等式可推广到任意斜形状的胞腔偏序集，其中 $\mathrm{bl}$ 被一个构造性定义的扩展 $\widetilde{\mathrm{bl}}$ 取代。我们证明了该猜想对每个 $2\times n$ 矩形成立——这是第一个真正的二维情形，其中块不再是路径区间，该统计量是两行的相关函数。我们引入一个双参数增长状态生成函数 $W_n(\alpha,\beta;t)$，通过底行最小值的秩 $\alpha$ 和最右顶胞块的块根的秩 $\beta$ 细化 $\sum_\sigma t^{\widetilde{\mathrm{bl}}(\sigma)}$，并证明闭式 $W_n(\alpha,\beta;t)=(\alpha-1)!\\,B(n,\beta-2)\\,S_n(\alpha,\beta;t)$（$n\ge 1$），其中 $B(n,k)$ 是 Catalan 三角形，$S_n$ 是上升阶乘骨架。证明是对 $n$ 通过 Kahane 的列插入转移进行归纳，其承重结构事实——新插入顶胞块下方的块根总是全局底行最小值——将归纳简化为 Catalan 三角形计算。对 $(\alpha,\beta)$ 求和得到望远镜和 $C_n\\,t^{\overline{n}}(t+1)^{\overline{n}}$，根据 MacMahon 乘积公式等于 $(2n)!\\,\Omega(R_{2,n};t)$。每个承重步骤都是完整的 $n$ 一般性论证；转移表和归纳步骤通过穷举枚举（直到 $n=8$ 和 $n=9$）使用独立的从零开始的实现进行交叉验证。一般斜形状和 $m>2$ 行仍待解决。
英文摘要
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.
Comments13 pages. Every load-bearing step is a complete general-n proof; the exhaustive computational cross-checks (transition table and inductive step enumerated via an independent from-scratch implementation, plus a Monte Carlo check beyond the brute-force ceiling) are honestly accounted for in Section 6. Not formally verified in a proof assistant. Methodology and AI disclosure in Section 7
