Volume and lattice points counting for the cyclopermutohedron

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 √ 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 V ol(K(L)).