On the equality of two generating functions related to plane partitions

Question

I'd like to prove $$\prod_{(i,j,k)\in\mathcal{B}(r,s,t)}\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}=\prod_{i=1}^r\prod_{j=1}^s\frac{1-q^{i+j+t-1}}{1-q^{i+j-1}},$$ where $$\mathcal{B}(r,s,t)=\{(i,j,k):1\leq i\leq r,1\leq j\leq s,1\leq k\leq t\}.$$ I'm guessing that we could find a bijection between products of terms from the left-hand side and terms from the right-hand side, but how I am not sure how. Of course there might be something easy/obvious that I am missing. Any help or general hints on how to approach problems like this would be appreciated!

Answer 1

The left hand side is the same as

$$\prod_{i=1}^r\prod_{j=1}^s\prod_{k=1}^t\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}$$

Now just consider

$$\prod_{k=1}^t\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}=\frac{1-q^{i+j+t-1}}{1-q^{i+j+1-2}}$$

because a lot of the factors cancel each other out.