The complexity of finite smooth words over binary alphabets

TL;DR

This paper investigates the structure and complexity of finite smooth words over binary alphabets, establishing key properties and making progress on a conjecture about their growth rate.

Contribution

It proves that f-smooth words are factors of smooth words and advances the understanding of their complexity growth, especially over even alphabets.

Findings

f-smooth words are exactly the factors of smooth words

proves the complexity growth conjecture over even alphabets

improves the upper bound for odd alphabets

Abstract

Smooth words over an alphabet of non-negative integers $\{a,b\}$ are infinite words that are infinitely derivable, the most famous example being the Oldenburger-Kolakoski word over $\{1,2\}$. The main way to study their language is to consider a finite version of smooth words that we call f-smooth words. In this paper we prove that the f-smooth words are exactly the factors of smooth words, and we make progress towards the conjecture of Sing that the complexity of f-smooth words over $\{a,b\}$ grows like $\Theta\left(n^{\log(a+b)/\log((a+b)/2)}\right)$: we prove it over even alphabets, we prove the lower bound over any binary alphabet and we improve the known upper bound over odd alphabets.