Permutations with Ascending and Descending Blocks

We investigate permutations in terms of their cycle structure and descent set. To do this, we generalize the classical bijection of Gessel and Reutenauer to deal with permutations that have some ascending and some descending blocks. We then provide the first bijective proofs of some known results. We also extend the work done in [4] by Eriksen, Freij, and Wastlund, who study derangements that descend in blocks of prescribed lengths. In particular, we solve some problems posed in [4] and also obtain a new combinatorial sum for counting derangements with ascending and descending blocks.