2015/05/02 by Ilya Nekrasov, Gaiane Panina, Nekrasov, Ilya +1 · 1 citation
Mathematics · #51M20 #52B20 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:51M20 #msc:52B20
paper · pdf · doi:10.48550/arxiv.1505.00352
arxiv created 2015/05/02 · arxiv updated 2015/05/05
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 √(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 K(L) related to the face lattice of cyclopermutohedron. Using this relationship, we introduce and compute the volume Vol(K(L)).