= Determinant formula for Specht module dimension
{title2=$f_\lambda=n!\det(1/(\lambda_i-i+j)!)$}
For a partition $\lambda\vdash n$ with $k$ positive parts, the number of <standard Young tableaux>, and hence the dimension of its complex <Specht module>, is $n!\det(1/(\lambda_i-i+j)!)$. Negative factorial arguments give zero entries. Setting $L_i=\lambda_i+k-i$ turns the determinant into $\prod_{i<j}(L_i-L_j)/\prod_iL_i!$ by the <Vandermonde determinant>, matching the <hook-length formula>.
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