Counterexamples to Stanley's conjecture on dimer coverings
Combinatorics
2026-05-28 v1
Abstract
Let be Stanley's explicit denominator for the dimer-covering generating function of rectangles. Stanley conjectured in 1985 that has only simple roots; this longstanding conjecture was recently recorded in Lai's list of open problems on tilings (see [6, Problem 33]). We disprove the conjecture by proving that and have repeated roots for every ; in particular, is the smallest counterexample. The construction comes from two exceptional multiplicative identities among trigonometric algebraic units. We further propose a conjecture concerning this class of trigonometric identities, which appears to be related to Robinson's problem on primitive Pell factors.
Cite
@article{arxiv.2605.28195,
title = {Counterexamples to Stanley's conjecture on dimer coverings},
author = {Xuejun Guo and Zhengyu Tao},
journal= {arXiv preprint arXiv:2605.28195},
year = {2026}
}
Comments
6 pages