For this example in Sommerville's book on calculating $r$-boundaries, i.e. a boundary of $r$ dimensions, and $N_r$ being defined as the number of $r$ boundaries. Similarly, defining $N_{pq}$ as the number of $p$-boundaries passing $(p<q)$ through $q$-boundary or likewise if $(p>q)$ lying in $q$-boundary. Here is the picture for the figure that these numbers are being calculated.
It is immediately seen from this figure that the number of points, lines and facets are $N_0=7, N_1=14, N_2=9$. Since this particular polyhedron is not in general configuration, the formula $N_q N_{pq} = N_p N_{qp}$ has to be modified to $\sum_{k} N^k_q N^k_{pq}=\sum_{k} N^k_p N^k_{qp}$ and that also the equations $\sum_k N^k_p=N_p$, where $N^k_p$ $p$-boundaries contain $N^k_{qp}$ $q$-boundaries for all $k$. This is what I have the question for.
The book says that $N_{01}=N_{21}=2$, which makes sense. It also says that $\sum_{k} N^k_2 N^k_{02}=8*3+1*4=28$, since it is defined that
$$ N_{02}=\left\{\begin{matrix} & 3 \mbox{ for } N_2=8\\ & 4 \mbox{ for } N_2=1 \end{matrix}\right. $$
Similarly, the further calculations are $\sum_{k} N^k_0 N^k_{20}=3+5*4+5=28$ and $\sum_{k} N^k_0 N^k_{10}=3+5*4+5=28$.
I am not sure how these are obtained since, for example for $N_{02}$ cases, the polyhedron has reduced edges so the calculation is done for a completely different figure, and i'm assuming it is only done for cases $N_2=1,8$ is because the region stays enclosed, since for cases of $N_2=2,...,7$ the region either does not enclose the region without having to merge edges or vertices or the number of vertices that define a facet becomes unequal for all facets.
For example in particular $N_2=8$ has a result which gives $N_{02}=\{3,3,3,3,3,3,3,4\}$ and when $N_2=1$ it is expected that from all the triangles in the figure that a single facet would have $3$ vertices, but instead it's shown that $N_{02}=4$ for $N_2=1$. Is there a particular way these numbers are subdivided and why they are divided this way? The book doesn't explain it in the definition. If someone could provide the explicit derivation or at least an explanation for $\sum_k N^k_{0} N^k_{20}$, $\sum_k N^k_{2} N^k_{20}$ and $\sum_k N^k_{0} N^k_{10}$ that would help.