Discretized configurations and partial partitions

TL;DR

This paper explores the topological structure of discretized configuration spaces in simplices, revealing their homotopy type and connecting them to combinatorial objects like Stirling numbers through generating functions.

Contribution

It establishes a homotopy equivalence for these configuration spaces and links their Euler characteristics to Stirling numbers via new combinatorial proofs.

Findings

Discretized configuration space is homotopy equivalent to a wedge of spheres.

The space is homeomorphic to the order complex of a poset of partial partitions.

Derived a topological proof of a recurrence relation for Stirling numbers.

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.