  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.
