 Beta-set hook-product identity

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.