The Number of Ribbon Tilings for Strips

First, we consider order-$n$ ribbon tilings of an $M$-by-$N$ rectangle $R_{M,N}$ where $M$ and $N$ are much larger than $n$. We prove the existence of the growth rate ${\gamma }_n$ of the number of tilings and show that ${\gamma }_n\le (n-1)\ln 2$. Then, we study a rectangle $R_{M,N}$ with fixed width $M=n$, called a strip. We derive lower and upper bounds on the growth rate ${\mu }_n$ for strips as $\ln n-1+o(1)\le {\mu }_n\le \ln n$. Besides, we construct a recursive system which enables us to enumerate the order-$n$ ribbon tilings of a strip for all $n\le 8$ and calculate the corresponding generating functions.