On the Number of Non-equivalent Parameterized Squares in a String
<|reference_start|>On the Number of Non-equivalent Parameterized Squares in a String: A string $s$ is called a parameterized square when $s = xy$ for strings $x$, $y$ and $x$ and $y$ are parameterized equivalent. Kociumaka et al. showed the number of parameterized squares, which are non-equivalent in parameterized equivalence, in a string of length $n$ that contains $\sigma$ distinct characters is at most $2 \sigma! n$ [TCS 2016]. In this paper, we show that the maximum number of non-equivalent parameterized squares is less than $\sigma n$, which significantly improves the best-known upper bound by Kociumaka et al.<|reference_end|>
@article{hamai2024on,
    title={On the Number of Non-equivalent Parameterized Squares in a String},
    author={Rikuya Hamai, Kazushi Taketsugu, Yuto Nakashima, Shunsuke Inenaga,
  Hideo Bannai},
    journal={arXiv preprint arXiv:2408.04920},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.04920},
    primaryClass={cs.DS cs.DM}
}