On the Number of Non-equivalent Parameterized Squares in a String

TL;DR

This paper improves the upper bound on the number of non-equivalent parameterized squares in a string from 2σ!n to less than σn, providing a tighter understanding of their maximum possible count.

Contribution

The paper presents a significantly improved upper bound on the maximum number of non-equivalent parameterized squares in a string, refining previous results.

Findings

Maximum number of non-equivalent parameterized squares is less than σn

Improved the upper bound from 2σ!n to σn

Provides tighter combinatorial bounds for parameterized squares

Abstract

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.