Consider the Vandermonde's determinant computed by
$$V(x_1,\dots,x_m):=\det(x_j^{i-1})_{i,j=1}^m=\prod_{1\leq i<j\leq m}(x_i-x_j).$$
The number of plane partitions in an $n\times m\times m$ box (MacMahon) is given by the neat product formula
$$\prod_{i=1}^n\prod_{j=1}^m\prod_{k=1}^m\frac{i+j+k-1}{i+j+k-2}.$$
Notation. Let $\mathcal{F}_m$ be the set of all $m$-element subsets of $[n+m]=\{1,2,\dots,n+m\}$. Write $\mathbf{J}=\{j_1,\dots,j_m\}$ for $\mathbf{J}\in\mathcal{F}_m$. The special element $\{1,2,\dots,m\}\in\mathcal{F}_m$ is denoted by $\mathbf{I}$, in which case $V(\mathbf{I})=(m-1)!!=1!\cdot2!\cdots(m-1)!$.
I would like to ask:
QUESTION. Is this expansion true? A combinatorial proof is desired, if possible.
$$\prod_{i=1}^n\prod_{j=1}^m\prod_{k=1}^m\frac{i+j+k-1}{i+j+k-2}
=\sum_{\mathbf{J}\in\mathcal{F}_m}\left(\frac{V(\mathbf{J})}{V(\mathbf{I})}\right)^2.$$
Remark. Observe the cute fact $\frac{V(\mathbf{J})}{V(\mathbf{I})}$ is always an integer (let's make Fedor's comment explicit).
Proof. Replacing $\mathbf{x}=(x_1,\dots,x_m)$ by $\mathbf{J}=(j_1,\dots,j_m)$ into
$$\det\left[\binom{x_j}i\right]
=\frac1{\prod_{i<j}(j-i)}\det\left[\prod_{k=0}^{i-1}(x_j-k)\right]
=\frac{\det\left[x_j^{i-1}\right]}{\prod_{i<j}(j-i)}
=\prod_{1\leq i \lt j\leq m} \frac{x_j-x_i}{j-i}.$$
results in integer entries, and thus integer determinant $\frac{V(\mathbf{J})}{V(\mathbf{I})}$. $\qquad\square$
Best Answer
I haven't worked out the details, but $V(\mathbf{J})/V(\mathbf{I})$ is the principal specialization of a Schur function. Then $\left( \frac{V(\mathbf{J})}{V(\mathbf{I})}\right)^2$ corresponds to a pair of SSYT (semistandard Young tableaux), which can be merged into a plane partition as in EC2, proof of Theorem 7.20.1. Summing over $\mathbf{J}$ should give all plane partitions fitting in an $n\times m\times m$ box.