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Let $\mathcal{P}$ be a simple thin polyomino and $\Bbbk$ a field. Let $R$ bethe toric $\Bbbk$-algebra associated to $\mathcal{P}$. Write the Hilbert seriesof $R$ as $h_{R}(t)/(1-t)^{\dim(R)}$. We show that$$(-1)^{\left\lfloor{\frac{\mathrm{deg} h_R(t)}{2}}\right\rfloor}h_{R}(-1) \geq0$$ if $R$ is Gorenstein. This shows that the Gorenstein rings associated tosimple thin polyominoes satisfy the Charney-Davis conjecture.

The Charney-Davis conjecture for simple thin polyominoes