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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Jonoska, N., Manea, F., Seki, S. (2014). A Stronger Square Conjecture on Binary Words. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds) SOFSEM 2014: Theory and Practice of Computer Science. SOFSEM 2014. Lecture Notes in Computer Science, vol 8327. Springer, Cham. https://doi.org/10.1007/978-3-319-04298-5_30
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DOI: https://doi.org/10.1007/978-3-319-04298-5_30
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-04297-8
Online ISBN: 978-3-319-04298-5
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