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Two words are $k$-binomially equivalent if each subword of length at most $k$occurs the same number of times in both words. The $k$-binomial complexity ofan infinite word is a counting function that maps $n$ to the number of$k$-binomial equivalence classes represented by its factors of length $n$.Cassaigne et al. [Int. J. Found. Comput. S., 22(4) (2011)] characterized afamily of morphisms, which we call Parikh-collinear, as those morphisms thatmap all words to words with bounded $1$-binomial complexity. Firstly, we extendthis characterization: they map words with bounded $k$-binomial complexity towords with bounded $(k+1)$-binomial complexity. As a consequence, fixed pointsof Parikh-collinear morphisms are shown to have bounded $k$-binomial complexityfor all $k$. Secondly, we give a new characterization of Sturmian words withrespect to their $k$-binomial complexity. Then we characterize recurrent wordshaving, for some $k$, the same $j$-binomial complexity as the Thue-Morse wordfor all $j\le k$. Finally, inspired by questions raised by Lejeune, we studythe relationships between the $k$- and $(k+1)$-binomial complexities ofinfinite words; as well as the link with the usual factor complexity.

Characterizations of families of morphisms and words via binomial complexities