The Number of Ribbon Tilings for Strips

TL;DR

This paper investigates the asymptotic number of ribbon tilings of large rectangles and strips, establishing growth rates, bounds, and recursive enumeration methods for small cases.

Contribution

It introduces the concept of growth rates for ribbon tilings, proves bounds for these rates, and develops recursive systems for enumeration of tilings in strips.

Findings

Existence of a growth rate b_n d; b1 (n-1) a0ln 2.

Bounds on strip growth rate b_mu_n 3d; b1 a0ln n.

Recursive systems for enumerating tilings for n a0a0 8.

Abstract

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.