The subword complexity of smooth words on 2-letter alphabets

TL;DR

This paper investigates the subword complexity of smooth words over 2-letter alphabets, establishing bounds based on the proportion of certain letters and extending previous results to more general cases.

Contribution

It provides new bounds on the number of smooth words of a given length, generalizing earlier findings and introducing conditions related to letter proportions.

Findings

Derived bounds for smooth words' counts based on letter proportion

Extended previous results to broader classes of alphabets

Established asymptotic behavior of subword complexity

Abstract

Let $\gamma_{a,b}(n)$ be the number of smooth words of length $n$ over the alphabet $\{a,b\}$ with $a<b$. Say that a smooth word $w$ is \emph{left fully extendable} (LFE) if both $aw$ and $bw$ are smooth. In this paper, we prove that for any positive number $\xi$ and positive integer $n_{0}$ such that the proportion of $b$'s is larger than $\xi$ for each LFE word of length exceeding $n_0$, there are two constants $c_{1}\,\textrm{and}\, c_{2}$ such that for each positive integer $n$, one has {eqnarray} c_{1}\cdot n^{\frac{\log (2b-1)}{\log (1+(a+b-2)(1-\xi))}}<\gamma_{a,b}(n)< c_2\cdot n^{\frac{\log (2b-1)}{\log (1+(a+b-2)\xi)}}. {eqnarray} In particular, taking $a=1\text{and}b=2$ in the above inequalities arrives at Huang and Weakley's result. Moreover, for 2-letter even alphabet $\{a,b\}$, there are two suitable constants $c_1,\,c_2$ such that \{eqnarray} c_{1}\cdot n^{\frac{\log (2b-1)}{\log ((a+b)/2)}}<\gamma_{a,b}(n)< c_2\cdot n^{\frac{\log (2b-1)}{\log ((a+b)/2)}}\textit{for each positive integer $n$}.\{eqnarray}