Subexponential upper bound on the number of rich words

TL;DR

The paper addresses the problem of bounding the number $R_q(n)$ of rich words of length $n$ over a finite alphabet. Building on the known subexponential growth $\lim_{n\to\infty}\sqrt[n]{R_q(n)}=1$, it develops two subexponential upper bounds: a recursive bound via a sequence $K_{\alpha}$ and an inductive scheme, and a nonrecursive closed-form bound using iterated logarithms. The main achievement is a closed-form bound $R(n)\le B(n)$ with $B(n)=q^{\frac{n}{\phi(n)}+\frac{n}{(2\lambda)^{f(n)-1}}}$ and $f(n)=\sqrt[\gamma]{c\ln^*(\frac{n}{\phi(n)}\ln q)}$, together with the result that $\lim_{n\to\infty}\sqrt[n]{B(n)}=1$. This provides an explicit subexponential upper bound for rich words, clarifying the growth rate and enabling sharper asymptotic analysis of $R_q(n)$. The work advances the understanding of palindromic structure in words and supplies practical tools for estimating the count of rich words in combinatorics on words.

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 $\frac{1}{2}<λ<1$ and $γ>1$ be real constants, let $q$ be the size of the alphabet, and let $φ$ be a positive function with $\lim_{n\rightarrow\infty}φ(n)=\infty$ and $\lim_{n\rightarrow\infty}\frac{n}{φ(n)}=\infty$. Let $\ln^*(x)$ denote the iterated logarithm of $x>0$. We prove that there are $n_0$ and $c>0$ such that if $n>n_0$, \[f(n)=\sqrt[γ]{c\ln^*{(\frac{n}{φ(n)}}\ln{q})}\quad\mbox{ and }\quad B(n)=q^{\frac{n}{φ(n)}+\frac{n}{(2λ)^{f(n)-1}}}\mbox{}\] then $\lim_{n\rightarrow\infty}\sqrt[n]{B(n)}=1$ and $R(n)\leq B(n)$.

Theorems & Definitions (30)

• Theorem 1
• Corollary 1
• proof
• Proposition 1
• Remark 1
• Remark 2
• Corollary 2
• proof
• Lemma 1
• proof