Classifying descents according to equivalence mod k

In an earlier paper the authors refine the well-known permutation statistic "descent" by fixing parity of (exactly) one of the descent's numbers. In the current paper, we generalize the results of that earlier paper by studying descents according to whether the first or the second element in a descent pair is divisible by k for some k≥2. We provide either an explicit or an inclusion-exclusion type formula for the distribution of the new statistics. Based on our results we obtain combinatorial proofs of a number of remarkable identities. We also provide bijective proofs of some of our results and state a number of open problems.