Subdivision of complexes of k-Trees

TL;DR

This paper proves that a certain poset's order complex is a subdivision of the complex of k-trees, extending nested set complex theory beyond lattices and answering a previously posed question.

Contribution

It introduces a novel generalization of nested set complexes to non-lattices, establishing a subdivision relationship between two complexes related to k-trees.

Findings

Order complex of a specific partition poset is a subdivision of k-trees complex.

Generalization of nested set complexes to non-lattices.

Answers a question posed by Feichtner.

Abstract

Consider the poset of partitions of {1,...(n-1)k+1} with block sizes congruent to 1 modulo k. We prove that its order complex is a subdivision of the complex of k-trees, thereby answering a question posed by Feichtner. The result is obtained by an ad-hoc generalization of concepts from the theory of nested set complexes to non-lattices.