Generalized Parikh mappings and homomorphisms

The notion of the Parikh mapping is generalized by considering numbers of occurrences of segments of a fixed length instead of considering numbers of letters (i.e., segments of length one) only as is done in connection with the Parikh mappings. It is easily seen that the families of regular and context-free languages make difference with respect to these generalized Parikh mappings. On the other hand, properties of the Parikh mappings in connection with λ-free homomorphisms are, in general, preserved in the generalization