English

On the number of squares in a finite word

Combinatorics 2022-04-27 v2

Abstract

A {\em square} is a word of the form uuuu. In this paper we prove that for a given finite word ww, the number of distinct square factors of ww is bounded by w\Alphabet(w)+1|w|-|\Alphabet(w)|+1, where w|w| denotes the length of ww and \Alphabet(w)|\Alphabet(w)| denotes the number of distinct letters in ww. This result answers a conjecture of Fraenkel and Simpson stated in 1998.

Keywords

Cite

@article{arxiv.2204.10204,
  title  = {On the number of squares in a finite word},
  author = {Srečko Brlek and Shuo Li},
  journal= {arXiv preprint arXiv:2204.10204},
  year   = {2022}
}