Abelian complexity function of the Tribonacci 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.

Revised version, 29 pages. Text rewritten, new results added (including results on the 4-bonacci word)

References in corpus (3)

• Abelian Complexity of Infinite Words Associated with Quadratic Parry Numbers
• Abelian complexity and Abelian co-decomposition
• Abelian properties of Parry words

Cited by in corpus (2)

• Abelian Combinatorics on Words: a Survey
• Abelian properties of Parry words