Distance in the Affine Buildings of SL_n and Sp_n

For a local field $K$ and $n\ge 2$, let ${\mathrm{\Xi }}_n$ and ${\mathrm{\Delta }}_n$ denote the affine buildings naturally associated to the special linear and symplectic groups ${\text{SL}}_n(K)$ and ${\text{Sp}}_n(K)$, respectively. We relate the number of vertices in ${\mathrm{\Xi }}_n$ ($n\ge 3$) close (i.e., gallery distance 1) to a given vertex in ${\mathrm{\Xi }}_n$ to the number of chambers in ${\mathrm{\Xi }}_n$ containing the given vertex, proving a conjecture of Schwartz and Shemanske. We then consider the special vertices in ${\mathrm{\Delta }}_n$ ($n\ge 2$) close to a given special vertex in ${\mathrm{\Delta }}_n$ (all the vertices in ${\mathrm{\Xi }}_n$ are special) and establish analogues of our results for ${\mathrm{\Delta }}_n$.