# Hook-length formula and applications to alternating permutations

## TL;DR

This paper explores applications of a recently proved hook-length formula, providing an alternative proof via continued fractions and extending the results to k-alternating permutations, linking them to weighted Dyck paths.

## Contribution

It offers a new proof of the hook-length formula's application to alternating permutations and generalizes the approach to k-alternating permutations.

## Key findings

- Established a non-trivial relation between alternating permutations and weighted Dyck paths.
- Provided an alternative proof using continued fractions.
- Extended the results to k-alternating permutations.

## Abstract

In this paper, we take interest in finding applications for a hook-length formula recently proved in (Morales Pak Panova 2016). This formula can be applied to give a non trivial relation between alternating permutations and weighted Dyck paths. First, we give an alternative proof for this result using continued fractions, and then we apply a similar reasoning to the more general case of $k$-alternating permutations.