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 \(\gamma_n \leq (n-1) \ln 2\). 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 \( \ln n - 1 + o(1) \leq \mu_n \leq \ln n \). Besides, we construct a recursive system which enables us to enumerate the order-\(n\) ribbon tilings of a strip for all \(n \leq 8\) and calculate the corresponding generating functions.