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{\bf Sergey Kitaev and Jeffrey Remmel}
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{\bf Classifying Descents According to Equivalence mod k}
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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
$k\geq 2$. 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.
\bye