Maximal Complexity of Finite Words

TL;DR

This paper investigates the maximal subword complexity of finite words, defining key measures like global maximal complexity and analyzing their properties and the number of words achieving these complexities.

Contribution

It introduces the concepts of maximal complexity and global maximal complexity for finite words, providing a framework for their analysis and enumeration.

Findings

Defined the maximal complexity C(w) for finite words.

Characterized the global maximal complexity K(N) for words of length N.

Analyzed the set R(N) of subword lengths achieving maximal complexity.

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.