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.

翻译：若一个非空真前缀同时是词~$w$ 的后缀，则称该前缀为 $w$ 的边界。若词~$w$ 的长度不超过 $1$，或者存在一个边界在 $w$ 中恰好出现两次，则称 $w$ 是闭的。若词~$w$ 的长度不超过 $1$，或者存在一个特权边界在 $w$ 中恰好出现两次，则称 $w$ 是特权的。令 $C_k(n)$（相应地，$P_k(n)$）表示 $k$ 字母表上长度为 $n$ 的闭词（相应地，特权词）的数量。本文改进了 $C_k(n)$ 与 $P_k(n)$ 的现有上下界。我们完全确定了 $C_k(n)$ 的渐近性态。对于 $P_k(n)$，我们通过给出一系列上下界几乎完全确定了其渐近性态，这些上下界之间的差距因子可以任意缓慢地增长。