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

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