Dyck Words, Pattern Avoidance, and Automatic Sequences

TL;DR

This paper explores Dyck words within binary sequences, characterizing factors of the Thue-Morse sequence, analyzing their properties, and establishing bounds on their counts, revealing insights into pattern avoidance and automatic sequences.

Contribution

It provides explicit characterizations of Dyck factors in Thue-Morse sequences and establishes bounds on their quantities, advancing understanding of pattern avoidance in automatic sequences.

Findings

Binary words that are 7/3-power-free have bounded nesting levels.

Explicit characterization of Dyck factors in Thue-Morse sequence.

Tight bounds on the number of Dyck factors of given length.

Abstract

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 $2n$.