Hook, line and sinker: a bijective proof of the skew shifted hook-length formula

TL;DR

This paper extends a hook-length formula to skew shifted shapes using a bijective proof, providing a unified and simplified combinatorial understanding of these formulas for both straight and shifted shapes.

Contribution

It introduces a bijective proof for the skew shifted hook-length formula, generalizing Naruse's formula and simplifying the understanding of shifted shape combinatorics.

Findings

Established a bijective proof for skew shifted hook-length formula

Unified the proof for straight and shifted shapes

Presented a weighted generalization of Naruse's formula

Abstract

A few years ago, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes, both straight and shifted. The formula involves a sum over objects called \emph{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. Recently, the formula for skew straight shapes was proved via a simple bumping algorithm. The aim of this paper is to extend this result to skew shifted shapes. Since straight skew shapes are special cases of skew shifted shapes, this is a bijection that proves the whole family of hook-length formulas, and is also the simplest known bijective proof for shifted (non-skew) shapes. A weighted generalization of Naruse's formula is also presented.