An umbral relation between pattern and commutation in strings

Two settings for string enumeration are considered in which string statistics can be constructed such that the generating series for the set of all strings have the form (F-r c a)’ in both cases, where F is a formal power series and u is a sequence. The two settings are qualitatively different. one involving pattern, which is locally testable, and the other involving commutation in strings, which is not locally testable. Evidence for a common generalization of these two settings is considered.