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

摘要

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 fc-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 ^--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 ψ~l(w) for an arbitrary morphism ψ. We also thoroughly describe the factors of the Thue-Morse word by introducing a relevant new equivalence relation.