Abstract
AbstractWe 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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