Asymptotic Bounds for the Number of Closed and Privileged Words

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 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 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.