Given the group $\mathrm{SL}(2,\mathbb{Q}\_p)$ one can describe precisely the number of vertices in the Bruhat-Tits building of distance $k$ from a fixed vertex $v$, where the distance between two vertices $v$ and $v'$ is the minimal number of edges in a path connecting the two. Denoting this quantity by $N\_k$, we see in particular that for $\mathrm{SL}(2,\mathbb{Q}\_p)$
$$N\_k=(q+1)q^{k-1},$$
where $q$ is the size of the residue field.


I am curious if one can come up with an analogous closed formula for $G=\mathrm{SL}(n,\mathbb{Q}\_p)$. In the case that $k=1$, this is not difficult and well known by using the lattice interpretation of the building for $\mathrm{SL}$ and counting the number of non-trivial $k$-dimensional subspaces of $\mathbb{F}\_p^n$ (one can also use spherical buildings for this and more general groups). For instance, in the case of $n=3$, we have
$$N\_1=2(q^2+q+1).$$
Continuing the example where $n=3$, to compute $N\_2$ is equivalent to finding the number of lattices $\Lambda$ such that if
$$\Lambda\_0=\mathbb{Z}\_p\oplus\mathbb{Z}\_p\oplus\mathbb{Z}\_p,$$
then
$$p^2\Lambda\_0\subsetneq \Lambda\subsetneq\Lambda\_0,$$
with the additional requirements that $\Lambda$ does not contain $p\Lambda\_0$ and that $\Lambda$ is not contained in $p\Lambda\_0$ (this ensures that $\Lambda$ is truly distance $2$). However, I have not been able to make the combinatorics of this count work on paper. In general when $n=3$, to compute $N\_k$ one would need to count the analogous $\Lambda$s in a chain with $p^2\Lambda\_0$ replaced by $p^k\Lambda\_0$ subject to the condition that $\Lambda$ does not contain any $p^i\Lambda\_0$ and is not contained in any $p^j\Lambda\_0$ where $1\le i,j\le k-1$ (to ensure genuine distance $k$).


I am looking to see if a closed formula for $N\_k$ is known in the literature or is easy to compute and I am missing the computation. I would be extremely pleased with just the case of $G=\mathrm{SL}(3,\mathbb{Q}\_p)$, even if just the case of $n=3$ and $p=2$.



Closed formula for vertices distance k in building of SLn
We consider the Bruhat-Tits building of $\text{SL}\_n(\mathbb{Q}\_p)$.
As usual, we identify the vertices with *lattices*, that is subgroups of the of $\mathbb{Q}^n$ commensurated with the *standard lattice* $\Lambda\_0=\mathbb{Z}\_p^n$, defined up to homothety, namely multiplication by $p^k$ for some $k\in \mathbb{Z}$.
Getting rid of the homothety relation, we can represent each lattice uniquely as a subgroup $\Lambda$ of $\Lambda\_0$ which is not contained in $p\Lambda\_0$.
Such a $\Lambda$ will contain $p^k\Lambda\_0$ for some natural $k$, and the minimal such $k$ is said to be the *distance* from $\Lambda$ to $\Lambda\_0$.


From now on by *vertex* I mean a representation of homothety class of lattices given as above.
Thus the set all vertices of distance $k$ from $\Lambda\_0$ is given by 
$$ \{\Lambda\mid p^k\Lambda\_0<\Lambda<\Lambda\_0,~\Lambda\notin p\Lambda\_0 \mbox{ and the rank of } p^k\Lambda\_0/\Lambda\mbox{ is less than }n\}.$$
Dividing by $p^k\Lambda\_0$, the set above is in bijection with the set of all $p$-abelian subgroups
$$ \{A \mid A < (\mathbb{Z}/p^k\mathbb{Z})^n,\mbox{ the ranks of } A\mbox{ and } (\mathbb{Z}/p^k\mathbb{Z})^n/A \mbox{ is less than }n\}.$$


Recall that every for every $p$-abelian group $A$ there exists a unique vector of natural numbers
$\lambda=(\lambda\_1,\lambda\_2,\ldots,\lambda\_m)$ with $\lambda\_1\geq \lambda\_2\geq \ldots\geq \lambda\_m> 0$ such that 
$$ A\simeq \bigoplus\_{i=1}^m \mathbb{Z}/p^{\lambda\_i}\mathbb{Z}.$$
In that case we say that $A$ is of (isomorphism) type $\lambda$.
If for all $i$ $\lambda\_i=k$ we say $\lambda=k^m$.


Given types $\nu$ and $\lambda$ we denote by ${\nu\choose \lambda}\_p$ the number of submodules of type $\lambda$ of a given module of type $\nu$. 
We conclude that the number of vertices of distance $k$ is given by $\sum {k^n\choose \lambda}\_p$ where the sum runs over all $\lambda=(\lambda\_1,\lambda\_2,\ldots,\lambda\_m)$ with $\lambda\_1=k$ and $m<n$.


We are left with a concrete computation.
Fixing $\nu$ and $\lambda$, ${\nu\choose \lambda}\_p$ is a polynomial in the variable $p$ which is computed in claim 4.5(1) in
<https://arxiv.org/pdf/math/0404408.pdf>:
$$ {\nu\choose \lambda}\_p=p^{\langle \nu'-\lambda',\lambda'\rangle}\prod\_i\begin{bmatrix} \nu'\_i-\lambda'\_{i+1} \\ \nu'\_i-\lambda'\_i\end{bmatrix}\_p$$
where $\nu',\lambda'$ are the transpose diagrams of $\nu,\lambda$, $\langle \nu,\lambda\rangle=\sum \nu\_i\lambda\_i$ and we use the notations:
$$ [i]\_1=1-p^{-i},~[m]\_p!=\prod\_{i=1}^m [i]\_p,~\begin{bmatrix} n \\ m\end{bmatrix}\_p =\frac{[n]\_p}{[m]\_p[n-m]\_p}.$$


