\[\sum_{\begin{subarray}{c}\pi\subseteq B(r,r,r)\\
\pi\mbox{\scriptsize\ cyclically symmetric}\end{subarray}}q^{\left|\pi\right|}%
=\prod_{h=1}^{r}\frac{1-q^{3h-1}}{1-q^{3h-2}}\prod_{1\leq h<j\leq r}\frac{1-q^%
{3(h+2j-1)}}{1-q^{3(h+j-1)}}=\prod_{h=1}^{r}\left(\frac{1-q^{3h-1}}{1-q^{3h-2}%
}\prod_{j=h}^{r}\frac{1-q^{3(r+h+j-1)}}{1-q^{3(2h+j-1)}}\right).\]