Classifying Descents According to equivalence mod k

In [S. Kitaev and J. Remmel: Classifying descents according to parity] the authors refine the well-known permutation statistic "descent" by fixing parity of (exactly) one of the descent's numbers. In this paper, we generalize the results of [S. Kitaev and J. Remmel: Classifying descents according to parity] by studying descents according to whether the first or the second element in a descent pair is equivalent to $k$ mod $k\geq 2$. 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.