MacMahon's enumeration of all [plane partions (PP)](https://en.wikipedia.org/wiki/Plane_partition) 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 in an $n$-cube
$${\tt SPP\_n}(q)=\prod\_{i,j=1}^n\frac{1-q^{i+j+n-1}}{1-q^{i+j+i-2}}.$$
The *ordinary* self-complementary plane partitions in a $(2n)$-cube
$${\tt SCPP\_{2n}}=\prod\_{i=1}^n \frac{(i-1)!^2 (i+2n-1)!^2}{(i+n-1)!^4}.$$
The following numerical observations fascinated me. Any explicit and pointed reference would be appreciated, if available.



> 
> **QUESTION.** Is there some sort of overarching explanation for these identities? Failing short of that, can you give a combinatorial or conceptual reason?
> $${\tt SPP\_{2n}(-1)=PP\_n(1)}, \qquad {\tt PP\_{2n}(-1)=SCPP\_{2n}}, \qquad
> {\tt PP\_{2n}(-1)=[PP\_n(1)]^2}.$$
> 
> 
> 


I believe that ${\tt PP\_{2n}(-1)=SCPP\_{2n}}$ is known.



$q$-plane partitions & specialization & interlinks
Stembridge's "$q=-1$ phenomenon" (the precursor to the cyclic sieving phenomenon) was developed precisely to explain these kind of evaluations of generating functions for plane partitions at $-1$. See Stembridge's ["Some hidden relations involving the ten symmetry classes of plane partitions"](https://www.sciencedirect.com/science/article/pii/0097316594901120) and ["On minuscule representations, plane partitions and involutions in complex Lie groups"](https://projecteuclid.org/euclid.dmj/1077288819). I haven't double-checked if all of the evaluations you mentioned are covered, but most of them should be.


EDIT: In response to comment below:


* Section 2.1 of "Some hidden relations..." proves the $PP\_{2n}(-1)=PP\_{n}(1)^2$ claim;
* Section 2.2 proves the $SPP\_{2n}(-1)=PP\_{n}(1)$ claim;
* The $PP\_{2n}(-1)=SCPP\_{2n}$ claim is a special case of Theorem 1.1, the main result of the paper.


EDIT 2:


In terms of conceptual explanations, the $PP\_{2n}(-1)=SCPP\_{2n}$ result has an explanation coming from Standard Monomial Theory of minuscule varieties (see the "On minuscule representations, ..." paper), but as far as I can tell the other evaluations you mentioned are just obtained by checking the product formulas in question.


EDIT 3 (much later):


Actually, the $SPP\_{2n}(-1)=PP\_{n}(1)$ result may have a "deeper" explanation. Namely, I think it follows from either the techniques in the Stembridge paper "On minuscule representations..." mentioned above or the techniques in Kuperberg's "Self-complementary plane partitions by Proctor's minuscule method" paper (<https://arxiv.org/abs/math/9411239>) (it depends on precisely which of the two $q$-analogs of symmetric plane partitions yours is, and I haven't checked that carefully) that $SPP\_{2n}(-1)$ should be the number of plane partitions in a $2n \times 2n \times 2n$ box which are both self-complementary and symmetric. This is known to be the same as the number of plane partitions in a $n\times n \times n$ box with no restrictions (see e.g. Case 7 of Stanley's classic survey <http://dedekind.mit.edu/~rstan/pubs/pubfiles/65.pdf>). There are even by now bijective proofs of this equality; see <https://arxiv.org/abs/1602.05535>.


