Upper bound for the number of privileged words

05/25/2022

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by   Josef Rukavicka, et al.

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A non-empty word w is a border of a word u if | w|<| u| and w is both a prefix and a suffix of u. A word u is privileged if | u|≤ 1 or if u has a privileged border w that appears exactly twice in u. Peltomäki (2016) presented the following open problem: “Give a nontrivial upper bound for B(n)”, where B(n) denotes the number of privileged words of length n. Let ln^[0](n)=n and let ln^[j](n)=ln(ln^[j-1](n)), where j,n are positive integers. We show that if q>1 is a size of the alphabet and j≥ 3 is an integer then there are constants α_j and n_j such that B(n)≤α_jq^n√(lnn)/√(n)ln^[j](n)∏_i=2^j-1√(ln^[i](n))n≥ n_j This result improves the upper bound of Rukavicka (2020).