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 dimen-sion 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 also compute the Euler characteristic in two different ways, thereby obtaining a topological proof of a combinatorial recurrence satisfied by the Stirling numbers of the second kind. 1