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Classifying descents according to equivalence mod k

Kitaev, Sergey and Remmel, Jeffrey (2006) Classifying descents according to equivalence mod k. In: Permutation Patterns Conference, 2006-06-12 - 2006-06-16, Reykjavik University.

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Abstract

In [5] the authors re¯ne the well-known permutation statistic \descent" by ¯xing parity of one of the descent's numbers. In this paper, we generalize the results of [5] by studying descents according to whether the ¯rst or the second element in a descent pair is equivalent to k mod 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.