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

Abstract

A \emph{power} is a word of the form $\underset{k\text{times}}{\underbrace{uu...u}}$, 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).