Volume and lattice points counting for the cyclopermutohedron

TL;DR

This paper explores the volume and lattice points of the cyclopermutohedron, linking its combinatorics to forests and configuration spaces, and provides formulas for these geometric and combinatorial properties.

Contribution

It establishes a connection between the cyclopermutohedron's volume, lattice points, and forest enumeration, extending known results from the permutohedron.

Findings

Volume of cyclopermutohedron relates to weighted forest counts

Derived a formula for the number of integer points in the cyclopermutohedron

Connected the configuration space of polygonal linkages to cyclopermutohedron face lattice

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