arXiv subjects
Publications and source records attributed to Matthew Zeckner.
We study the simplicial complex that arises from non-attacking rook placements on a subclass of Ferrers boards that have $a_i$ rows of length $i$ where $a_i>0$ and $i\leq n$ for some positive integer $n$. In particular, we will investigate enumerative properties of their facets, their homotopy type, and homology.
In 2003, A. Bjorner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SG_{n,k} is homotopy equivalent to a k-sphere. Further, for n=2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SG_{n,k} contains as a deformation retract the boundary complex of a simplicial polytope. Our purpose is to give a positive answer to this question in the case k=2. We also find in this case that, after partially subdividing the neighborhood complex, the resulting complex deformation retracts onto a subcomplex arising as a polyhedral boundary sphere that is invariant under the action induced by the automorphism group of SG_{n,2}.