On the abelian complexity of generalized Thue-Morse sequences

TL;DR

This paper determines the exact abelian complexity of generalized Thue-Morse sequences for all lengths and shows their complexity functions are ultimately periodic and k-automatic, revealing structural regularities.

Contribution

It provides explicit formulas for abelian complexity of generalized Thue-Morse sequences and proves their automaticity and periodicity properties.

Findings

Exact abelian complexity values for all n ≥ k

Complexity functions are ultimately periodic with period k

Abelian complexities are k-automatic 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 $n\geq k$. 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.