Complexes of trees and nested set complexes

Details Complexes of trees and nested set complexes Complexes of trees and nested set complexes

2004-09-14

arxiv.org/abs/math/0409235v2

Mathematics

Combinatorics

13 pages, 5 figures; minor revision, references updated, to appear in Pacific J. Math

Scientific paper

We exhibit an identity of abstract simplicial complexes between the well-studied complex of trees and the reduced minimal nested set complex of the partition lattice. We conclude that the order complex of the partition lattice can be obtained from the complex of trees by a sequence of stellar subdivisions. We provide an explicit cohomology basis for the complex of trees that emerges naturally from this context. Motivated by these results, we review the generalization of complexes of trees to complexes of $k$-trees by Hanlon, and we propose yet another, in the context of nested set complexes more natural, generalization.

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