On the 2-binomial complexity of the generalized Thue–Morse words

Abstract

In this paper, we study the 2-binomial complexity btm,2(n) of the generalized Thue–Morse words tm over the alphabet {0,1,…,m−1} for every integer m≥3. By using boundary words, we fully characterize when two factors of tm are 2-binomially equivalent. In particular, we obtain the exact value of btm,2(n) for every integer n≥m2. As a consequence, btm,2(n) is ultimately periodic with period m2. This result partially answers a question of Lejeune et al. (2020) [11].