Two-dimensional Fibonacci Words: Tandem Repeats and Factor Complexity

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.