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