Upper bound for the number of closed and privileged words

TL;DR

This paper establishes an upper bound on the number of closed and privileged words of a given length over an alphabet, addressing an open problem by providing a logarithmic factor times the main exponential growth term.

Contribution

It introduces a new upper bound for the number of closed and privileged words, advancing understanding of their combinatorial properties.

Findings

Upper bound for closed words: $D(n) \\leq c \\ln n \\frac{q^{n}}{\\sqrt{n}}$

Privileged words are a subset of closed words, with an implied upper bound

Addresses an open problem posed by Peltomäki (2016)

Abstract

A non-empty word $w$ is a border of the word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ with the border $w$ is closed if $u$ has exactly two occurrences of $w$. A word $u$ is privileged if $\vert u\vert\leq 1$ or if $u$ contains a privileged border $w$ that appears exactly twice in $u$. Peltom\"aki (2016) presented the following open problem: "Give a nontrivial upper bound for $B(n)$", where $B(n)$ denotes the number of privileged words of length $n$. Let $D(n)$ denote the number of closed words of length $n$. Let $q>1$ be the size of the alphabet. We show that there is a positive real constant $c$ such that \[D(n)\leq c\ln{n}\frac{q^{n}}{\sqrt{n}}\mbox{, where }n>1\mbox{.}\] Privileged words are a subset of closed words, hence we show also an upper bound for the number of privileged words.