Binomial Complexities and Parikh-Collinear Morphisms

Inspired by questions raised by Lejeune, we study the relationships between the k and (k + 1)-binomial complexities of an infinite word; 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. We further study binomial properties of the images of aperiodic binary words in general, and Sturmian words in particular, by a power of the Thue-Morse morphism.