Subexponential upper bound on the number of rich words

TL;DR

This paper establishes a subexponential upper bound on the number of rich words of length n over a finite alphabet, improving understanding of their growth rate and confirming it is slower than exponential.

Contribution

It provides the first explicit subexponential upper bound on R(n), the count of rich words, using novel analytical techniques involving iterated logarithms.

Findings

R(n) grows subexponentially with n

Established explicit upper bounds involving iterated logarithms

Confirmed the limit of the nth root of R(n) is 1

Abstract

Let $R(n)$ denote the number of rich words of length $n$ over a given finite alphabet. In 2017 it was proved that $\lim_{n\rightarrow\infty} \sqrt[n]{R(n)}=1$; it means the number of rich words has a subexponential growth. However, up to now, no subexponential upper bound on $R(n)$ has been presented. The current paper fills this gap. Let $\frac{1}{2}<\lambda<1$ and $\gamma>1$ be real constants, let $q$ be the size of the alphabet, and let $\phi$ be a positive function with $\lim_{n\rightarrow\infty}\phi(n)=\infty$ and $\lim_{n\rightarrow\infty}\frac{n}{\phi(n)}=\infty$. Let $\ln^*(x)$ denote the iterated logarithm of $x>0$. We prove that there are $n_0$ and $c>0$ such that if $n>n_0$, \[f(n)=\sqrt[\gamma]{c\ln^*{(\frac{n}{\phi(n)}}\ln{q})}\quad\mbox{ and }\quad B(n)=q^{\frac{n}{\phi(n)}+\frac{n}{(2\lambda)^{f(n)-1}}}\mbox{}\] then $\lim_{n\rightarrow\infty}\sqrt[n]{B(n)}=1$ and $R(n)\leq B(n)$.