Stanley’s hook content formula

The number of semi-standard Young tableaux $Θ$ of shape $𝜆$ with all $Θ (𝑖, 𝑗) \leq 𝑁$ is given by

𝑚 𝑁 𝜆 = \prod 𝑖, 𝑗 \in 𝜆 𝑁 - 𝑖 + 𝑗 ℎ 𝜆 (𝑖, 𝑗)

where $ℎ 𝜆 (𝑖, 𝑗)$ is the Hook length. comb

Proof

proof Very complicated combinatorial proof