Maximal Complexity of Finite Words

Abstract

The subword complexity of a finite word $w$ of length $N$ is a function which associates to each $n\le N$ the number of all distinct subwords of $w$ having the length $n$. We define the \emph{maximal complexity} C(w) as the maximum of the subword complexity for $n\in \{1,2,...,N\}$, and the \emph{global maximal complexity} K(N) as the maximum of C(w) for all words $w$ of a fixed length $N$ over a finite alphabet. By R(N) we will denote the set of the values $i$ for which there exits a word of length $N$ having K(N) subwords of length $i$. M(N) represents the number of words of length $N$ whose maximal complexity is equal to the global maximal complexity.