On column-convex and convex Carlitz polyominoes

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}