English

Counterexamples to Stanley's conjecture on dimer coverings

Combinatorics 2026-05-28 v1

Abstract

Let Qk(x)Q_k(x) be Stanley's explicit denominator for the dimer-covering generating function Fk(x)=n0Ak,nxnF_k(x)=\sum_{n\ge0}A_{k,n}x^n of k×nk\times n rectangles. Stanley conjectured in 1985 that Qk(x)Q_k(x) 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 Q14h1(x)Q_{14h-1}(x) and Q30h1(x)Q_{30h-1}(x) have repeated roots for every h1h\ge1; in particular, k=13k=13 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

R2 v1 2026-07-22T07:36:44.083Z