The complexity of finite smooth words over binary 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 $Θ\left(n^{\log(a+b)/\log((a+b)/2)}\right)$: 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.

Figures (1)

• Figure 1: The tree $T$ over $\{1,2\}$.

Theorems & Definitions (59)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.1
• Definition 1.5
• Example 1.1
• Definition 1.6
• Example 1.2
• Proposition 1.1