# A bijective proof of the hook-length formula for skew shapes

## TL;DR

This paper provides a new bijective proof of Naruse's hook-length formula for skew shapes, extending classical bijections to more complex combinatorial objects and offering a novel proof of the classical hook-length formula.

## Contribution

It introduces a simple bijection that proves an equivalent recursive version of Naruse's formula and offers a new bijective proof of the classical hook-length formula.

## Key findings

- Established a bijective proof of Naruse's hook-length formula for skew shapes.
- Derived a new bijective proof of the classical hook-length formula.
- Connected the proof techniques for skew shapes with classical results.

## Abstract

Recently, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes. The formula involves a sum over objects called excited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall. In this paper, we present a simple bijection that proves an equivalent recursive version of Naruse's result, in the same way that the celebrated hook-walk proof due to Green, Nijenhuis and Wilf gives a bijective (or probabilistic) proof of the hook-length formula for ordinary shapes. In particular, we also give a new bijective proof of the classical hook-length formula, quite different from the known proofs.