# “quantum” symmetric plane partitions beget alternating sign matrices?

The "quantum" version qTSPP of the number of totally symmetric plane partitions, contained in the cube $[0,n]^3$, is enumerated by $$f_n(q):=\prod_{j=1}^n\prod_{k=1}^j\prod_{\ell=1}^k\frac{1-q^{j+k+\ell-1}}{1-q^{j+k+\ell-2}}.$$ L'Hopital $f_n(1)=\lim_{q\rightarrow1}f_n(q)$ restores the classical version $\prod_{1\leq\ell\leq k\leq j\leq n}\frac{j+k+\ell-1}{j+k+\ell-2}$. Although $f_n(-1)=0$ trivially, when $n$ is odd, I observe the case $n$ even is decidedly striking; namely that, $$f_{2n}(-1)=\lim_{q\rightarrow -1}f_{2n}(q)=\prod_{k=0}^{n-1}\frac{(3k+1)!}{(n+k)!},$$ the number $A_n$ of $n\times n$ Alternating Sign Matrices or $ASMs$.

QUESTION still waiting for an answer.

Is there a non-analytic (more conceptual) reason for this connection between qTSPP and ASMs?

• Maybe it is related to this question? mathoverflow.net/questions/247965/… – Per Alexandersson Dec 6 '16 at 0:28
• Your conjecture is a nice example of Stembridge's $q=-1$ phenomenon, not that this observation helps with a proof. – Richard Stanley Dec 6 '16 at 2:00
• Curiosity asking: Is there even such a thing as "quantum symmetric plane partitions" known? I understand that $f_n(q)$ is a natural way to quantize the generating series, but do we know a statistic on symmetric plane partitions that the yields this series as generating function? (Sorry if this is something well-known.) – darij grinberg Dec 6 '16 at 8:09
• Instead of $\sum_{\pi}1$, you do the weighted sum $\sum_{\pi}q^{\vert\pi\vert}$. – T. Amdeberhan Dec 6 '16 at 14:36
• @darijgrinberg: see arxiv.org/abs/1002.4384 – Sam Hopkins Dec 6 '16 at 23:11

This is not really an answer to the question per se (in that it is not conceptual), but I wanted to point out that this $$q=-1$$ evaluation was essentially already observed in Stembridge's original paper about the $$q=-1$$ phenomenon (as hinted at by the comment of Richard Stanley). Indeed, the point of that paper is that if $$G$$ is a subgroup of $$S_3$$ acting on plane partitions, and $$P'_G(q)$$ is the "appropriate" $$q$$-analog generating function for the plane partitions invariant under $$G$$, then the number of plane partitions invariant under $$G$$ and also self-complementary is $$P'_G(-1)$$. A particular case is $$G=S_3$$, which then says that the evaluation of the $$q$$-analog of the number of totally symmetric plane partitions at $$q=-1$$ is the number of totally symmetric, self-complementary plane partitions. You ask about alternating sign matrices rather than TSSCPPs, but of course the number of ASMs and TSSCPPs is the same.