  In this paper, we introduce and study {\it Carlitz polyominoes}. In
particular, we show that, as $n$ grows to infinity, asymptotically the number
of \begin{enumerate} \item column-convex Carlitz polyominoes with perimeter
$2n$ is \beq \frac{9\sqrt{2}(14+3\sqrt{3})}{2704\sqrt{\pi n^3}}4^n. \feq \item
convex Carlitz polyominoes with perimeter $2n$ is \beq
\frac{n+1}{10}\left(\frac{3+\sqrt{5}}{2}\right)^{n-2}. \feq \end{enumerate}
