The weighted hook length formula

TL;DR

This paper introduces a weighted version of the hook length formula, providing two proofs—bijective and probabilistic—and explores further applications in combinatorics.

Contribution

It presents the first weighted analogue of the branching rule for the hook length formula with two distinct proofs and additional applications.

Findings

A new weighted hook length formula is established.

Two proofs: bijective and probabilistic, are provided.

Applications extend the classical hook length formula framework.

Abstract

Based on the ideas in [CKP], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk proof of Green-Nijenhuis-Wilf, as well as the q-walk of Kerov. Further applications are also presented.