Two-dimensional Fibonacci Words: Tandem Repeats and Factor Complexity

TL;DR

This paper investigates tandem repeats and factor complexity in two-dimensional Fibonacci arrays, deriving exact counts and generating methods, thereby advancing understanding of their combinatorial properties.

Contribution

It provides an explicit formula for the number of tandem repeats in finite Fibonacci arrays and develops methods to generate these arrays using two-dimensional homomorphisms.

Findings

Exact number of tandems in finite Fibonacci arrays derived

Factor complexities of finite and infinite Fibonacci arrays computed

Generation of Fibonacci arrays via two-dimensional homomorphism achieved

Abstract

If $x$ is a non-empty string then the repetition $xx$ is called a tandem repeat. Similarly, a tandem in a two dimensional array $X$ is a configuration consisting of a same primitive block $W$ that touch each other with one side or corner. In \cite{Apostolico:2000}, Apostolico and Brimkov have proved various bounds for the number of tandems in a two dimensional word of size $m \times n$. Of the two types of tandems considered therein, they also proved that, for one type, the number of occurrences in an $m \times n$ Fibonacci array attained the general upper bound, $\mathcal{O}(m^{2}n \hspace{0.1cm} \mbox{log} \hspace{0.1cm} n)$. In this paper, we derive an expression for the exact number of tandems in a given finite Fibonacci array $f_{m,n}$. As a required result, we derive the factor complexities of $f_{m,n}$, $m,n \ge 0$ and that of the infinite Fibonacci word $f_{\infty, \infty}$. Generations of $f_{\infty, \infty}$ and $f_{m,n}$, for any given $m,n \ge 1$ using a two-dimensional homomorphism is also achieved.