Two 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 the 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 ${\mathrm{\Psi }}^{\ell }(w)$ for an arbitrary morphism Ψ. We also thoroughly describe the factors of the Thue–Morse word by introducing a relevant new equivalence relation.

只要两个词共享相同的子词（即子序列），且子序列的长度最多为k且具有相同的多重性，则它们在k二项式上是等效的。这是阿贝尔对等和西蒙全等的改进。无穷词x的k二项式复杂度通过此k二项式等价关系将整数n映射到x中出现的长度为n的因子集的商中的类数。这种复杂性的度量方法尚未得到足够的研究。在本文中，我们描述了k-摩尔斯单词的二项式复杂度。与更熟悉的复杂度函数相比，结果令人震惊。尽管Thue-Morse单词是非周期性的，但其k二项式复杂度最终仅取两个值。在本文中，我们首先表示以迭代形式出现的子词出现的次数${\mathrm{\Psi }}^{\ell }（w）$an 通过引入相关的新的等价关系，我们还彻底描述了Thue-Morse单词的因素。