Templates for the k-Binomial Complexity of the Tribonacci Word

Consider k-binomial equivalence: two finite words are equivalent if they share the same subwords of length at most k with the same multiplicities. With this relation, the k-binomial complexity of an infinite word x maps the integer n to the number of pairwise non-equivalent factors of length n occurring in x. In this paper based on the notion of template introduced by Currie et al., we show that, for all k >= 2, the k-binomial complexity of the Tribonacci word coincides with its usual factor complexity p(n) = 2n + 1. A similar result was already known for Sturmian words, but the proof relies on completely different techniques that seemingly could not be applied for Tribonacci.