# Identity for classes of plane partitions

There are several classes of plane partitions in the literature.

Among these, let's look at the enumeration of three of them: the symmetric (SPP), totally symmetric (TSPP) and totally symmetric and self-complementary plane partitions (TSSCPP), given respectively, by \begin{align*} {\tt SPP}_n&=\prod_{i,j=1}^n\frac{i+j+n-1}{i+j+j-2}, \\ {\tt TSPP}_n&=\prod_{1\leq i\leq j\leq n}\frac{i+j+n-1}{i+j+j-2}, \\ {\tt TSSCPP}_{2n}&=\prod_{1\leq i\leq j\leq n}\frac{i+j+n-1}{i+j+i-1}. \end{align*}

I know how to prove the below result, algebraically.

QUESTION. Can you provide a combinatorial or conceptual reason for this equality? $${\tt SPP_n=TSPP_n\times TSSCPP_{2n}}.$$

• This is also true at the level of q-analogues. – F. C. Mar 4 at 20:28
• That is true, yes. – T. Amdeberhan Mar 4 at 20:50