Upper bound for the number of closed and privileged words

11/25/2019
by   Josef Rukavicka, et al.
0

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|≤ 1 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)≤ clnnq^n/√(n)n>1 Privileged words are a subset of closed words, hence we show also an upper bound for the number of privileged words.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset