A closed form for the block-statistic generating function of the $2\times n$ rectangle: the two-row case of Kahane's conjecture

Abstract

Kahane (arXiv:2607.11225) proves that for a fence poset $P$ of size $n$, $n!\Omega (P;t)={\sum }_{\sigma \in S_n}{t}^{\mathrm{bl}(\sigma )}$, with $\mathrm{bl}(\sigma )$ 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 $W_n(\alpha ,\beta ;t)$ refining ${\sum }_{\sigma }{t}^{\widetilde{\mathrm{bl}}(\sigma )}$ by the rank $\alpha$ of the bottom-row minimum and the rank $\beta$ of the rightmost top cell's block-root, and prove the closed form $W_n(\alpha ,\beta ;t)=(\alpha -1)!B(n,\beta -2)S_n(\alpha ,\beta ;t)$ ($n\ge 1$), 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 $(\alpha ,\beta )$ telescopes to $C_n{t}^{\overline{n}}(t+1{)}^{\overline{n}}$, which by MacMahon's product formula equals $(2n)!\Omega (R_{2,n};t)$. 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.

arXiv comment13 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