Sum of squares of hook lengths and contents

It is known that for the Young diagram of any partition of n, the sum of squares of the hook lengths of its cells is exactly n² more than that sum of squares of the contents of its cells. That is, for any λ n, {equation*} ∑u ∈ λ h(u)^2 = n^2 + ∑u ∈ λ c(u)^2. {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 higher powers of hook lengths and contents.