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arXiv · 1505.00352

Volume and lattice points counting for the cyclopermutohedron

Ilya Nekrasov·Gaiane Panina
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Abstract

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 $Vol(\mathcal{K}(L))$.

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Ilya Nekrasov, Gaiane Panina. 2015-05-02. Volume and lattice points counting for the cyclopermutohedron. https://arxiv.org/abs/1505.00352

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