On column-convex and convex Carlitz polyominoes

TL;DR

This paper introduces Carlitz polyominoes, analyzing their asymptotic enumeration for column-convex and convex types with respect to perimeter, providing explicit formulas for their growth rates as size increases.

Contribution

It defines Carlitz polyominoes and derives asymptotic formulas for the number of such polyominoes with given perimeter, a novel enumeration in polyomino theory.

Findings

Asymptotic count for column-convex Carlitz polyominoes with perimeter 2n.

Asymptotic count for convex Carlitz polyominoes with perimeter 2n.

Explicit formulas involving exponential growth rates.

Abstract

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}