Discretized configurations and partial partitions

Abstract

We show that the discretized configuration space of $k$ points in the $n$-simplex is homotopy equivalent to a wedge of spheres of dimension $n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions of $\{1,...,n+1\}$ with exactly $k$ parts. We compute the exponential generating function for the Euler characteristic of this space in two different ways, thereby obtaining a topological proof of a combinatorial recurrence satisfied by the Stirling numbers of the second kind.