Improved estimates for the number of privileged words

TL;DR

This paper improves the lower bounds on the number of privileged words of length n over an alphabet of size q, providing more accurate estimates for large n and generalizing previous results.

Contribution

It introduces new asymptotic bounds for privileged words, extending prior work to all alphabet sizes and large n using combinatorial and analytical methods.

Findings

Established a lower bound of (cq^n)/(n(log_q n)^2) for privileged words

Generalized previous bounds to all alphabet sizes q

Provided asymptotic estimates for large n

Abstract

In combinatorics on words, a word w of length n over an alphabet of size q is said to be privileged if n <= 1 or if n >= 2 and w has a privileged border that occurs exactly twice in w. Forsyth, Jayakumar and Shallit proved that there exist at least 2^{n-5}/n^2 privileged binary words of length n. Using the work of Guibas and Odlyzko, we prove that there are constants c and n_0 such that for n >= n_0, there are at least (cq^n)/(n(\log_q n)^2) privileged words of length n over an alphabet of size q. Thus, for n sufficiently large, we improve the earlier bound set by Forsyth, Jayakumar and Shallit and generalize for all q.