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Two-dimensional k-abelian complexity of the Tribonacci word

Prove that the two-dimensional sequence (k,n) ↦ #1{k}{t}(n), where t is the Tribonacci fixed point of the substitution 0→01, 1→02, 2→0, is not synchronized, and demonstrate that it is computed by a sequence automaton whose recurrence polynomial equals (X−1)(X^3−X^2−X−1).

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Background

The authors give extensive new results about the Tribonacci word, including a linear representation for its two-dimensional generalized abelian complexity and automaticity of certain discrete derivatives. Despite this progress, the precise global characterization of the two-dimensional function remains unresolved.

They conjecture a negative answer for synchronization and propose that the function is nonetheless generated by a sequence automaton with recurrence polynomial equal to the product of (X−1) and the minimal polynomial of the Tribonacci substitution (X3−X2−X−1).

References

Let ${t}$ be the Tribonacci sequence, fixed point of $0\mapsto 01, 1\mapsto 02, 2 \mapsto 0$. The $2$-dimensional sequence $(#1{k}{t}(n))_{k\ge 1, n\ge 0}$ is not synchronized but computed by a sequence automaton of polynomial $(X-1)(X3-X2-X-1)$.

— Effective Computation of Generalized Abelian Complexity for Pisot Type Substitutive Sequences (2504.13584 - Couvreur et al., 18 Apr 2025) in Section 7 (Open problems and questions)