• Corpus ID: 118366413

Volume and lattice points counting for the cyclopermutohedron

@article{Nekrasov2015VolumeAL,
  title={Volume and lattice points counting for the cyclopermutohedron},
  author={I. I. Nekrasov and Gaiane Panina},
  journal={arXiv: Metric Geometry},
  year={2015}
}

• I. Nekrasov, G. Panina
• Published 2 May 2015
• Mathematics
• arXiv: Metric Geometry

The face lattice of the permutohedron realizes the combinatorics of linearly ordered partitions of the set $[n]=\{1,...,n\}$. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of $[n]$.
It is known that the volume of the standard permutohedron equals the number of trees with $n$ labeled vertices multiplied by $\sqrt{n}$. The number of integer points of the standard permutohedron equals the number of forests on $n$ labeled vertices.
In the paper we prove that the volume of the cyclopermutohedron also equals some weighted number of forests, which eventually reduces to zero. We also derive a combinatorial formula for the number of integer points in the cyclopermutohedron.
Another object of the paper is the configuration space of a polygonal linkage $L$. It has a cell decomposition $\mathcal{K}(L)$ related to the face lattice of cyclopermutohedron. Using this relationship, we introduce and compute the volume $Vol(\mathcal{K}(L))$.