MacMahon's enumeration of all plane partions (PP) 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.


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" and "On minuscule representations, plane partitions and involutions in complex Lie groups". 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.


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.

  • $\begingroup$ Sam, I was aware this paper you mentioned. However, I do not see all of them. Maybe some. $\endgroup$ – T. Amdeberhan Mar 5 at 23:04
  • $\begingroup$ @T.Amdeberhan: I think they're all in the "Some hidden relations..." paper. See the edit. $\endgroup$ – Sam Hopkins Mar 5 at 23:24
  • $\begingroup$ Thank you, I'll take a closer look. $\endgroup$ – T. Amdeberhan Mar 5 at 23:26

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