In this paper we investigate upper bounds for the length of the preprojective partition for selfinjective algebras. The preprojective partition has been defined for arbitrary Artin algebras by Auslander and Smalo in [ 2). For a representation-finite algebra, that is, an algebra II for which the set indn of isomorphism classes of finitely generated indecomposable Amodules is finite, the preprojective partition can be defined as the unique decomposition of ind II into disjoint non-empty sets P,,, P, ,..., P,(,,, obtained inductively as follows: Pi is the smallest subset of Qi = ind/i\(P, U P, u ..* U Pi,) such that for every module M in an isomorphism class in Qi there is an epimorphism from a finite direct sum of representatives of isomorphism classes in Pi onto M. The preinjective partition indA = I,U II u -.a LJIi(*, with 1, # 0 is defined dually. D. Zacharia proved that p(A) and $4) coincide if n is a hereditary representation-finite algebra and that p(A) + 1 equals the maximum of the lengths of the indecomposable A-modules. If the graph underlying the quiver of /i is the Dynkin-diagram A,, D,, E,, E,, or E,, then p(A) equals n 1, 2n 4, 10, 16, or 28, respectively (9 1. We will prove the following theorem: