= Beta-set hook-product identity
{title2=$H_\lambda=\prod_i\ell_i!/\prod_{i<j}(\ell_i-\ell_j)$}
For $\ell_i=\lambda_i+m-i$, the hook lengths in row $i$ are $\ell_i-q$, where $q$ runs over the non-beta numbers in $[0,\ell_i-1]$. Their product is $\ell_i!/\prod_{k>i}(\ell_i-\ell_k)$. Multiplying over rows proves the identity and converts the <hook-length formula> into a <Vandermonde determinant> divided by factorials. Padding the partition with zero rows changes the beta set but not the hook product.
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