Bialternant formula
= Bialternant formula
{title2=$s_\lambda=a_{\lambda+\delta}/a_\delta$}
Let $a_\gamma=\det[x_j^{\gamma_i}]$ and $\delta=(N-1,\ldots,0)$. Multiply the <Jacobi–Trudi identity> <matrix> by $B_{jk}=(-1)^{N-j}e_{N-j}(X\setminus x_k)$. The generating-function identity $H(t)\prod_{i\ne k}(1-x_it)=(1-x_kt)^{-1}$ makes the product <matrix> $[x_k^{\lambda_i+N-i}]$. Setting $\lambda=0$ shows $\det B=a_\delta$, proving the formula as a <polynomial> identity.