Discretized configurations and partial partitions

Abrams, Aaron; Gay, David; Hower, Valerie

doi:10.1090/S0002-9939-2012-10816-0

Discretized configurations and partial partitions
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by Aaron Abrams, David Gay and Valerie Hower PDF

Proc. Amer. Math. Soc. 141 (2013), 1093-1104 Request permission

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 $\{1,\dots ,n+1\}$ 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.

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