Binomial Complexities and Parikh-Collinear Morphisms

Two words are $k$-binomially equivalent, if each word of length at most $k$ occurs as a subword, or scattered factor, the same number of times in both words. The $k$-binomial complexity of an infinite word maps the natural $n$ to the number of $k$-binomial equivalence classes represented by its factors of length $n$. Inspired by questions raised by Lejeune, we study the relationships between the $k$ and $(k+1)$-binomial complexities; as well as the link with the usual factor complexity. We show that pure morphic words obtained by iterating a Parikh-collinear morphism, i.e. a morphism mapping all words to words with bounded abelian complexity, have bounded $k$-binomial complexity. In particular, we study the properties of the image of a Sturmian word by an iterate of the Thue-Morse morphism.