The question is here.

For convenience:

Let $S=M_n(\mathbb{Z}_2)$. Let $D^{(n)}_2$ be the matrices in the rank $2$ $\mathcal{D}$-class of $S$. Find

$$N_n=\left\lvert\left\{\begin{array} \, e & \mathcal{L} & f \\ \mathcal{R} & \, & \mathcal{R} \\ h & \mathcal{L} & g \end{array}\mid e, f, g, h\in E\left(D^{(n)}_2\right)\right\}\right\rvert;$$ that is, find $N_n$, the number of quadruples $(e, f, g,h)\in E\left(D_2^{(n)}\right)^4$ such that $e\mathcal{L}f\mathcal{R}g\mathcal{L}h\mathcal{R}e$.