Derangements with Ascending and Descending Blocks

TL;DR

This paper generalizes the study of derangements with prescribed ascending and descending blocks, providing a generating function and a new combinatorial sum for counting such derangements, thus extending prior work.

Contribution

It introduces a unified framework for derangements with mixed ascending and descending blocks, including a generating function and combinatorial interpretations.

Findings

Derived a generating function for derangements with mixed blocks

Established a new combinatorial sum for counting these derangements

Extended previous work on derangements with descending blocks

Abstract

We continue the work of Eriksen, Freij, and Wastlund [3], who study derangements that descend in blocks of prescribed lengths. We generalize their work to derangements that ascend in some blocks and descend in others. In particular, we obtain a generating function for the derangements that ascend in blocks of prescribed lengths, thus solving a problem posed in [3]. We also work towards a combinatorial interpretation of a polynomial sum appearing in [3]. As a result, we obtain a new combinatorial sum for counting derangements with ascending and descending blocks.