A note on the maximum number of $k$-powers in a finite word

TL;DR

This paper establishes bounds on the maximum number of distinct $k$-power factors in finite words, generalizing previous results on squares, and provides tight asymptotic estimates for these quantities.

Contribution

It derives new bounds for the maximum number of distinct $k$-power factors in words, extending known results on squares to higher powers and different exponents.

Findings

Maximum number of $k$-power factors is between $rac{n}{k-1}- ext{Theta}( oot{n} ext{)}$ and $rac{n-1}{k-1}$.

Maximum number of power factors with exponent at least 2 is at most $n-1$.

Bounds generalize recent upper bounds on square factors in words.

Abstract

A \emph{power} is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer; the power is also called a {\em $k$-power} and $k$ is its {\em exponent}. We prove that for any $k \ge 2$, the maximum number of different non-empty $k$-power factors in a word of length $n$ is between $\frac{n}{k-1}-\Theta(\sqrt{n})$ and $\frac{n-1}{k-1}$. We also show that the maximum number of different non-empty power factors of exponent at least 2 in a length-$n$ word is at most $n-1$. Both upper bounds generalize the recent upper bound of $n-1$ on the maximum number of different square factors in a length-$n$ word by Brlek and Li (2022).