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- W224063989 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 $Vol(mathcal{K}(L))$." @default.
- W224063989 created "2016-06-24" @default.
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- W224063989 date "2015-05-02" @default.
- W224063989 modified "2023-09-27" @default.
- W224063989 title "Volume and lattice points counting for the cyclopermutohedron" @default.
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