Polytopal enumeration of circular extensions for arbitrary partial cyclic orders

Determine whether every partial cyclic order admits a pair of integral polytopes whose equal normalized volumes enumerate its circular extensions.

Background

The paper places the partial cyclic orders Z′_G within a broader program that uses integral polytopes to enumerate circular extensions through normalized volumes. Existing constructions apply to particular classes of partial cyclic orders, while the authors note that a general construction is unresolved. The partial cyclic orders arising from connected downward closed graphs are proposed as new test cases, with consecutive coordinate polytopes suggested as a possible candidate.