On a class of languages with holonomic generating functions

We define a class of languages (RCM) obtained by considering Regular languages, linear Constraints on the number of occurrences of symbols and Morphisms. The class RCM presents some interesting closure properties, and contains languages with holonomic generating functions. As a matter of fact, RCM is related to one-way 1-reversal bounded k-counter machines and also to Parikh automata on letters. Indeed, RCM is contained in LNFCM but not in LDFCM, and strictly includes LLPA. We conjecture that LDFCM⊂RCM.