A closed form for the block-statistic generating function of the 2× 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! Ω(P;t)=∑σ∈ Sn tbl(σ), with bl(σ) a canonical block count, and conjectures (Conjecture 4.4) that the identity extends to the cell poset of any skew shape, with bl replaced by a constructively-defined extension bl̃. We prove the conjecture for every 2× 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 Wn(α,β;t) refining ∑σtbl̃(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 Wn(α,β;t)=(α−1)! B(n,β−2) Sn(α,β;t) (n≥ 1), where B(n,k) is the Catalan triangle and Sn 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 Cn tn̅(t+1)n̅, which by MacMahon's product formula equals (2n)! Ω(R2,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.
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BibTeX
@misc{white2026closed,
title = {{A closed form for the block-statistic generating function of the $2\times n$ rectangle: the two-row case of Kahane's conjecture}},
author = {Patrick White},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17561},
eprint = {2609.17561},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17561},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A closed form for the block-statistic generating function of the 2× n rectangle: the two-row case of Kahane's conjecture
AU - Patrick White
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.17561
UR - https://arxiv.org/abs/2609.17561
AB - Kahane (arXiv:2607.11225) proves that for a fence poset P of size n, n! Ω(P;t)=∑_σ∈ S_n t^bl(σ), with bl(σ) a canonical block count, and conjectures (Conjecture 4.4) that the identity extends to the cell poset of any skew shape, with bl replaced by a constructively-defined extension bl̃. We prove the conjecture for every 2× 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 ∑_σt^bl̃(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≥ 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^n̅(t+1)^n̅, 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.
ER -