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

TL;DR

This paper determines the exact 2-binomial complexity of generalized Thue-Morse words for all sufficiently large lengths, showing it is ultimately periodic with a period related to the defining parameter m.

Contribution

It provides the exact values of 2-binomial complexity for generalized Thue-Morse words and proves its periodicity, addressing an open question in the field.

Findings

Exact 2-binomial complexity values for n ≥ m^2

Complexity is ultimately periodic with period m^2

Partially answers a question by Lejeune, Leroy, and Rigo

Abstract

In this paper, we study the $2$-binomial complexity $b_{\mathbf{t}_{m},2}(n)$ of the generalized Thue-Morse words $\mathbf{t}_{m}$ for every integer $m\geq 3$. We obtain the exact value of $b_{\mathbf{t}_{m},2}(n)$ for every integer $n\geq m^{2}$. As a consequence, $b_{\mathbf{t}_{m},2}(n)$ is ultimately periodic with period $m^{2}$. This result partially answers a question of M. Lejeune, J. Leroy and M. Rigo [Computing the $k$-binomial complexity of the Thue-Morse word, J. Comb. Theory Ser. A, {\bf 176} (2020) 105284].