= Solution
The <Young diagram> is $[\lambda]=\{(i,j):1\leq i\leq k,\ 1\leq j\leq\lambda_i\}$. The <hook length> of a box is
$$
h(i,j)=\lambda_i-j+\lambda'_j-i+1,
$$
counting the box itself, the boxes to its right in the same row, and the boxes below it in the same column. The <hook-length formula> is
$$
\boxed{f^\lambda=\frac{n!}{\prod_{(i,j)\in[\lambda]}h(i,j)}.}
$$
The first implication in the equivalence is immediate: \b[if every <hook length> is odd, no hook can have length two].
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