Stanley's distinct-roots conjecture for dimer-covering denominators

Let $k$ be a positive integer, let $A_{k,n}$ denote the number of domino tilings of a $k\times n$ rectangle, and let

$$F_k(x)=\sum_{n\geq 0}A_{k,n}x^n.$$

Stanley's denominator $Q_k(x)\in\mathbb{Z}[x]$ is defined by $F_k(x)=P_k(x)/Q_k(x)$ and $Q_k(0)=1$; explicitly, with $\ell=\left\lfloor k/2\right\rfloor$, set

$$c_j=\cos\frac{j\pi}{k+1}+\sqrt{1+\cos^2\frac{j\pi}{k+1}},\qquad \bar{c}_j=\cos\frac{j\pi}{k+1}-\sqrt{1+\cos^2\frac{j\pi}{k+1}},$$

and, for $S\subseteq\{1,\ldots,\ell\}$,

$$a_S=\left(\prod_{j\in S}c_j\right)\left(\prod_{j\in\{1,\ldots,\ell\}\setminus S}\bar{c}_j\right).$$

Then

$$Q_k(x)= \begin{cases} \displaystyle\prod_{S\subseteq\{1,\ldots,\ell\}}(1-a_Sx),&k\text{ even},\\[1.2em] \displaystyle\prod_{S\subseteq\{1,\ldots,\ell\}}(1-a_S^2x^2),&k\text{ odd}. \end{cases}$$

Stanley's conjecture. The polynomial $Q_k(x)$ has distinct roots. This conjecture concerns the simplicity of the poles of the dimer-covering generating functions. It is refuted: the paper proves that $Q_{14h-1}(x)$ and $Q_{30h-1}(x)$ have repeated roots for every $h\geq1$, with $k=13$ the smallest counterexample.