On the abelian complexity of generalized Thue-Morse sequences

Abstract

In this paper, we study the abelian complexity ${\rho }_n^{ab}({\mathbf{t}}^{(k)})$ of generalized Thue-Morse sequences ${\mathbf{t}}^{(k)}$. We obtain the exact value of ${\rho }_n^{ab}({\mathbf{t}}^{(k)})$ for every integer $n\ge k$. Consequently, ${\rho }_n^{ab}({\mathbf{t}}^{(k)})$ is ultimately periodic with the period $k$. Moreover, we show that the abelian complexities of a class of infinite sequences are $k$-automatic.