= Vandermonde shift identity
{c}
{title2=$\sum_i x_i\Delta(x+te_i)=(\sum_i x_i+\binom m2t)\Delta(x)$}
The left side is an <alternating polynomial> of total degree one more than $\Delta(x)$. Dividing by the <Vandermonde determinant> gives a symmetric homogeneous <polynomial> of degree one, necessarily $a\sum_i x_i+bt$. At $t=0$, $a=1$; differentiation in $t$ and the <Euler theorem for homogeneous functions> give $b=\binom m2$. Thus the identity holds as a <polynomial> identity, even at repeated coordinates. At shifted partition coordinates with $t=-1$, it proves the removable-corner recurrence in the <hook-length formula>.
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