Asymptotic bounds for the number of closed and privileged words

TL;DR

This work analyzes the asymptotic counts of closed and privileged words over a $k$-letter alphabet. It completely resolves the growth of the number of closed words, proving $C_k(n)=\Theta\left(\frac{k^n}{n}\right)$, and provides a near-complete picture for privileged words by establishing a hierarchy of upper and lower bounds that differ by factors shrinking with iterated logarithms. The approach combines border-avoidance and border-binding analyses via $A_k(n,u)$ and $B_k(n,u)$, together with auto-correlation theory (Guibas–Odlyzko) to tightly control contributions from borders. The results advance understanding of combinatorial word structures and connect to prefix-synchronized codes, with implications for coding theory and the study of periodic-like and rich word families.

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.

Theorems & Definitions (32)

• Example 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6: Nicholson and Rampersad Nicholson2018
• proof : Proof of Theorem \ref{['theorem:mainC']} \ref{['theorem:mainCLower']}
• Lemma 7