Dyck Words, Pattern Avoidance, and Automatic Sequences

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 $7\mathrm{/}3$-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 $2n$.