Counting profile strings from rectangular tilings

Abstract

We study the enumeration of profiles, which are outlines that occur when tiling a rectangular board with squares, dominoes, and trominoes. Profiles of length $m$ correspond to a special subset of the set ${\left\{0,1,2,3\right\}}^m$, called profile strings. Profiles and their corresponding strings first appeared in the enumeration of the tilings of rectangular $2\times n$ and $3\times n$ boards with squares, dominoes, and trominoes. Profiles also play a role in enumerating the tilings of an $m\times n$ board for any fixed $m\ge 2$. We describe how profiles arise when enumerating tilings, and we prove that the number of profile strings of length $m$ equals $m\cdot 3^{m-1}$.