Distance in the Affine Buildings of SLn and Spn

For a local field K and n ≥ 2, let Ξn and ∆n denote the affine buildings naturally associated to the special linear and symplectic groups SLn(K) and Spn(K), respectively. We relate the number of vertices in Ξn (n ≥ 3) close (i.e., gallery distance 1) to a given vertex in Ξn to the number of chambers in Ξn containing the given vertex, proving a conjecture of Schwartz and Shemanske. We then consider the special vertices in ∆n (n ≥ 2) close to a given special vertex in ∆n (all the vertices in Ξn are special) and establish analogues of our results for ∆n.