Counting inversion sequences by parity successions and runs

Let I_n denote the set of inversion sequences of length n. In this paper, we enumerate members of I_n represented using positive integers according to parameters which track the number of runs and successions of letters of a given parity. To this end, we consider a four-parameter joint distribution on I_n and derive several general results for this distribution, providing further treatment of some particular cases. In order to prove our results, we make use of a non-standard generating function which allows for the system of recurrences satisfied by various refinements of the aforementioned joint distribution to be translated into a system of linear differential equations that can be solved explicitly. Finally, the more general problem of enumerating the members of I_n according to adjacencies involving two elements that are equivalent modulo m is considered.