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

qTSPPandASMs?