Hook length formula

The number of standard Young tableaux in the shape of a given Young diagram $𝜆$ with $𝑛$ boxes is given by comb

𝑑 𝜆 = \prod 𝑖, 𝑗 \in 𝜆 𝑛! ℎ 𝜆 (𝑖, 𝑗)

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

Proof

proof Very complicated combinatorial proof

Example

Using Further notation we may write for $𝜆 = (4, 3, 1)$

𝑑 𝜆 = ∣ 87654321 ∣ ∣ 64314211 ∣ = 70