Permutations with Ascending and Descending Blocks

TL;DR

This paper generalizes a classical permutation bijection to permutations with mixed ascending and descending blocks, providing new bijective proofs and solving open problems related to block-descending derangements.

Contribution

It introduces a generalized bijection for permutations with ascending and descending blocks and offers the first bijective proofs of certain known results, addressing open problems.

Findings

Generalized bijection for permutations with mixed blocks

First bijective proofs of known results in this area

Solved open problems on block-descending derangements

Abstract

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 solve some problems posed in [3] by Eriksen, Freij, and Wastlund, who study derangements that descend in blocks of prescribed lengths.