On the shape of subword complexity sequences of finite words

TL;DR

This paper investigates the properties and shape of subword complexity sequences of finite words, analyzing how various factors influence their structure and enumerating possible sequences for different word lengths and alphabets.

Contribution

It provides new insights into the properties of subword complexity sequences and estimates their counts for various word lengths and alphabet sizes.

Findings

Characterization of subword complexity sequence properties

Enumeration of distinct sequences for specific lengths and alphabets

Conjectures on the growth rate of the number of sequences

Abstract

The subword complexity of a word $w$ over a finite alphabet $\mathcal{A}$ is a function that assigns for each positive integer $n$, the number of distinct subwords of length $n$ in $w$. The subword complexity of a word is a good measure of the randomness of the word and gives insight to what the word itself looks like. In this paper, we discuss the properties of subword complexity sequences, and consider different variables that influence their shape. We also compute the number of distinct subword complexity sequences for certain lengths of words over different alphabets, and state some conjectures about the growth of these numbers.