It is required to show that
$$ s_\lambda(q,q^2,\cdots,q^m) = q^{m(\lambda)}\prod_{i,j \in \lambda}\frac{1-q^{c_{i,j}+m}}{1-q^{h_{i,j}}} $$
where $c_{i,j}=j-i$ is the content of cell $(i,j)$, and $h_{i,j}$ is the hook length, $m(\lambda) = \sum_i i\lambda_i$, and $s_\lambda$ is the Schur function.
Seriously I spent several days and found no proof on the internet at all. People say this is Stanley's formula, and I found the original paper (Theory and Application of Plane Partitions. Part 2), and there is no clue. (Symbols in that paper are vastly different from what we use today, but if I'm right he only did the case when $q=1$).
What I thought was (correct me at any chance), since $s_\lambda(x_1,x_2,\cdots,x_m)$ is the generating function of all semistandard $\lambda$-tableaux with non-negative entries of size less than $m$. If we replace $x_i$ with $q^i$, then I can combine all terms accordingly. I was thinking about a way to 'scan' the tableaux cell-by-cell, which may envolve hook length and content. But I didn't know how to start.