Cyclopermutohedron

Abstract: It is known that the $k$-faces of the permutohedron $\Pi_n$ are labeled by (all possible) linearly ordered partitions of the set $[n]={1,...,n}$ into $(n-k)$ non-empty parts. The incidence relation corresponds to the refinement: a face $F$ contains a face $F'$ whenever the label of $F'$ refines the label of $F$. In the paper we consider the cell complex ${CP}$ defined in analogous way, replacing linear ordering by cyclic ordering. Namely, $k$-cells of the complex ${CP}$ are labeled by (all possible) cyclically ordered partitions of the set $[n+1]={1,...,n, n+1}$ into $(n+1-k)$ non-empty parts, where $(n+1-k)>2$. The incidence relation again corresponds to the refinement: a cell $F$ contains a cell $F'$ whenever the label of $F'$ refines the label of $F$. In particular, two vertices are joined by an edge whenever their labels differ on a permutation of two neighbor elements. The complex ${CP}$ cannot be represented by a convex polytope, since it is not a combinatorial sphere (not even a combinatorial manifold). However, it can be represented by some \textit{virtual polytope} (Minkowski difference of two convex polytopes) which we call "cyclopermutohedron" $\mathcal{CP}_{n+1}$. It is defined explicitly, as a weighted Minkowski sum of line segments. Informally, the cyclopermutohedron can be viewed as "permutohedron with diagonals". One of the motivations is that the cyclopermutohedron is a "universal" polytope for moduli spaces of polygonal linkages.