A Simple Proof of the Hook Length Formula

Abstract

In this note, we give a simple and direct proof for the Hook Length Formula. The simplicity of our proof relies on the usage of the residue theorem as a short cut. The number of standard Young tableaux for a given Ferrers diagram . is the dimension of the irreducible representation of the symmetric group corresponding to X. The hook length formula (which was first proved in [1]) is a method for calculating this number and is a surprisingly beautiful formula (because the problem looks complex but the formula looks naive). Let us begin by giving the precise definitions of Ferrers diagrams, standard Young tableaux, and the hook length formula. Suppose that N and m are positive integers and that k = (kI, ..., Xm) is a sequence