# A Stronger Square Conjecture on Binary Words

• Nataša Jonoska
• Florin Manea
• Shinnosuke Seki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8327)

## Abstract

We propose a stronger conjecture regarding the number of distinct squares in a binary word. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a word is upper bounded by the length of the word. Here, we conjecture that in the case of a word of length n over the alphabet {a,b}, the number of distinct squares is upper bounded by $$\frac{2k-1}{2k+2}n$$, where k is the least of the number of a’s and the number of b’s. We support the conjecture by showing its validity for several classes of binary words. We also prove that the bound is tight.

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© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Nataša Jonoska
• 1
• Florin Manea
• 2
• Shinnosuke Seki
• 3
• 4
1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
2. 2.Institut für InformatikChristian-Albrechts-Universität zu KielKielGermany
3. 3.Helsinki Institute for Information Technology (HIIT)Finland
4. 4.Department of Information and Computer ScienceAalto UniversityAaltoFinland