Dyck Words, Pattern Avoidance, and Automatic Sequences

01/15/2023

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by   Lucas Mol, et al.

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We study various aspects of Dyck words appearing in binary sequences, where 0 is treated as a left parenthesis and 1 as a right parenthesis. We show that binary words that are 7/3-power-free have bounded nesting level, but this no longer holds for larger repetition exponents. We give an explicit characterization of the factors of the Thue-Morse word that are Dyck, and show how to count them. We also prove tight upper and lower bounds on f(n), the number of Dyck factors of Thue-Morse of length 2n.