Sum of squares of hook lengths and contents

Abstract

It is known that for the Young diagram of any partition of an integer $n$, the sum of squares of the hook lengths of its cells is exactly $n^2$ more than that of the contents of its cells. That is, for any partition $\lambda$ of an integer $n$, \begin{equation*} \sum_{u \in \lambda} h(u)^2 = n^2 + \sum_{u \in \lambda} c(u)^2. \end{equation*} We provide a bijective proof of this fact, thus solving a problem posed by Stanley. Along the way, we obtain a formula for the number of rectangles in the Young diagram of a partition. We also mention a result for sums of other powers of hook lengths and contents.