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.