Upper bound for the number of closed and privileged words

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F20%3A00345707" target="_blank" >RIV/68407700:21340/20:00345707 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1016/j.ipl.2020.105917" target="_blank" >https://doi.org/10.1016/j.ipl.2020.105917</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.ipl.2020.105917" target="_blank" >10.1016/j.ipl.2020.105917</a>

Result language
angličtina
Original language name
Upper bound for the number of closed and privileged words
Original language description
A non-empty word w is a border of the word u if |w|<|u| and w is both a prefix and a suffix of u. A word u with the border w is closed if u has exactly two occurrences of w. A word u is privileged if |u|<2 or if u contains 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 D(n) denote the number of closed words of length n. Let q>1 be the size of the alphabet. We show that there is a positive real constant c such that D(n) <= (c.ln(n).q^n)/(sqrt(n)), where n>1. Privileged words are a subset of closed words, hence we show also an upper bound for the number of privileged words.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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