Abelian complexity function of the Tribonacci word

TL;DR

This paper develops an automaton to efficiently compute the abelian complexity of the Tribonacci word, revealing its automatic sequence nature and solving an open problem about specific complexity values.

Contribution

It introduces a fast automaton-based method to evaluate the abelian complexity of the Tribonacci and 4-bonacci words, characterizing when certain complexity values occur.

Findings

Automaton evaluates abelian complexity in O(log n) operations.

Characterization of n for which the complexity equals specific values.

Extension of approach to the 4-bonacci word.

Abstract

According to a result of Richomme, Saari and Zamboni, the abelian complexity of the Tribonacci word satisfies $\rho^{\mathrm{ab}}(n)\in\{3,4,5,6,7\}$ for each $n\in\mathbb{N}$. In this paper we derive an automaton that evaluates the function $\rho^{\mathrm{ab}}(n)$ explicitly. The automaton takes the Tribonacci representation of $n$ as its input; therefore, $(\rho^{\mathrm{ab}}(n))_{n\in\mathbb{N}}$ is an automatic sequence in a generalized sense. Since our evaluation of $\rho^{\mathrm{ab}}(n)$ uses $\mathcal{O}(\log n)$ operations, it is fast even for large values of $n$. Our result also leads to a solution of an open problem proposed by Richomme et al. concerning the characterization of those $n$ for which $\rho^{\mathrm{ab}}(n)=c$ with $c$ belonging to $\{4,5,6,7\}$. In addition, we apply the same approach on the $4$-bonacci word. In this way we find a description of the abelian complexity of the $4$-bonacci word, too.