Stirling complexes

TL;DR

This paper explores the topological structure of resource distribution spaces on trees, revealing they are homotopy equivalent to wedges of spheres and depend only on resource and node counts.

Contribution

It introduces Stirling complexes as cubical complexes modeling resource distributions and proves their homotopy equivalence to wedges of spheres, independent of tree structure.

Findings

Stirling complexes are homotopy equivalent to wedges of spheres.

The homotopy type depends only on resource and node counts.

Provides combinatorial formulas for counting spheres.

Abstract

In this paper we study natural reconfiguration spaces associated to the problem of distributing a fixed number of resources to labeled nodes of a tree network, so that no node is left empty. These spaces turn out to be cubical complexes, which can be thought of as higher-dimensional geometric extensions of the combinatorial Stirling problem of partitioning a set of named objects into non-empty labeled parts. As our main result, we prove that these Stirling complexes are always homotopy equivalent to wedges of spheres of the same dimension. Furthermore, we provide several combinatorial formulae to count these spheres. Somewhat surprisingly, the homotopy type of the Stirling complexes turns out to depend only on the number of resources and the number of the labeled nodes, not on the actual structure of the tree network.