论文标题
有限词的最大$ k $能力的注释
A note on the maximum number of $k$-powers in a finite word
论文作者
论文摘要
a \ emph {power}是$ \ UnderBrace {uu ... u} _ {k \; \ text {times}} $，其中$ u $是一个词，$ k $是一个正整数；该电源也称为{\ em $ k $ -power}，$ k $是其{\ em endent}。我们证明，对于任何$ k \ ge 2 $，在长度$ n $中的不同非空的$ k $ - 功率因子的最大数量介于$ \ frac {n} {k-1} {k-1}-θ（\ sqrt {n}）$和$ \ frac {n-1} {k-1} {k-1} $之间。我们还表明，指数的最大数量至少为2 $ n $ word，最多是$ n-1 $。两种上限都将$ n-1 $的最近上限概括为Brlek and Li（2022）的长度 - $ n $ Word的不同正方形因子的最大数量。
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 $\frac{n}{k-1}-Θ(\sqrt{n})$ 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).
