Asymptotic bounds for the number of closed and privileged words

TL;DR

This paper investigates the asymptotic growth of the number of closed and privileged words over finite alphabets, providing precise bounds and resolving their asymptotic behaviors.

Contribution

It improves bounds on the counts of closed and privileged words and fully characterizes the asymptotic behavior of closed words, nearly doing so for privileged words.

Findings

Resolved the asymptotic behavior of closed words.

Provided bounds for privileged words separated by a slowly growing factor.

Improved existing upper and lower bounds on the counts.

Abstract

A word~$w$ has a border $u$ if $u$ is a non-empty proper prefix and suffix of $u$. A word~$w$ is said to be \emph{closed} if $w$ is of length at most $1$ or if $w$ has a border that occurs exactly twice in $w$. A word~$w$ is said to be \emph{privileged} if $w$ is of length at most $1$ or if $w$ has a privileged border that occurs exactly twice in $w$. Let $C_k(n)$ (resp.~$P_k(n)$) be the number of length-$n$ closed (resp. privileged) words over a $k$-letter alphabet. In this paper, we improve existing upper and lower bounds on $C_k(n)$ and $P_k(n)$. We completely resolve the asymptotic behaviour of $C_k(n)$. We also nearly completely resolve the asymptotic behaviour of $P_k(n)$ by giving a family of upper and lower bounds that are separated by a factor that grows arbitrarily slowly.