Computing the k-binomial Complexity of the Thue-Morse Word

Two finite words are k-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most k with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The k-binomial complexity of an infinite word x maps the integer n to the number of classes in the quotient, by this k-binomial equivalence relation, of the set of factors of length n occurring in x. This complexity measure has not been investigated very much. In this paper, we characterize the k-binomial complexity of the Thue-Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue-Morse word is aperiodic, its k-binomial complexity eventually takes only two values. In this paper, we first express the number of occurrences of subwords appearing in iterates of the form Psi(l)(w) for an arbitrary morphism Psi. We also thoroughly describe the factors of the Thue-Morse word by introducing a relevant new equivalence relation.