Hook inequalities

TL;DR

This paper provides an elementary proof of a hook inequality involving Young diagram hooks and introduces numerous similar inequalities and high-dimensional generalizations.

Contribution

It offers a simple proof of a recent hook inequality and extends it to a broad class of related inequalities and high-dimensional cases.

Findings

Elementary proof of the hook inequality

Derivation of various similar inequalities

High-dimensional generalizations of hook inequalities

Abstract

We give an elementary proof of the recent hook inequality given in [MPP3]: $\prod_{u\in \lambda} h(u) \, \le \, \prod_{u\in \lambda} h^\ast(u),$ where $h(u)$ is the usual hook in Young diagram $\lambda$, and $h^\ast(i,j)=i+j-1$. We then obtain a large variety of similar inequalities and their high-dimensional generalizations.