An Elementary Proof of the Hook Formula

Abstract

The hook-length formula is a well known result expressing the number of standard tableaux of shape $\lambda$ in terms of the lengths of the hooks in the diagram of $\lambda$. Many proofs of this fact have been given, of varying complexity. We present here an elementary new proof which uses nothing more than the fundamental theorem of algebra. This proof was suggested by a $q,t$-analog of the hook formula given by Garsia and Tesler, and is roughly based on the inductive approach of Greene, Nijenhuis and Wilf. We also prove the hook formula in the case of shifted Young tableaux using the same technique.