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MacMahon's enumeration of all plane partions (PP) inside an $n$-cube generalizes to $${\tt PP_n}(q)=\prod_{i,j,k=1}^n\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}.$$ A $q$-analogue of symmetric plane partitions ...

Consider the standard Vandermonde $V(x_1, \ldots, x_n) = \prod_{i < j} (x_i - x_j)$. I am intersted in the calculation of the following expression for fixed $k$: $$\sum_i (x_i)^k (d/dx_i)^k V(x_1 , ...