Dosen and Petric, following earlier work of Devadoss and others, have used hypergraphs to describe a class of polytopes lying between the permutohedron and the simplex, obtained by successive truncations of faces of the latter (at all dimensions). The key notion is that of constructs, which are combinatorial structures derived from the structure of the hypergraph under study and describe all the faces of an abstract polytope. We present a handy notation for them.
Just like associahedra arise in the coherence of monoidal categories, several polytopes (the associahedron, the permutohedron, and a third one, called hemi-associahedron) intervene in the investigation of the notion of weak Cat-operad (in which the two kinds of operadic associativities - the sequential and the parallel one - are weakened to be coherent isomorphisms). The situation for weak cyclic operads is currently under investigation.