Figure 25 on pg. of "Cataland: Why the Fuss?" by Christian Stump, Hugh Thomas, and Nathan Williams depicts cluster complexes (CCs) associated with generalized $(m)$-Narayana / noncrossing partitions / parking functions, etc. I have partition polynomials (ParPs) which give the number of facets of these CCs as the absolute value of the coefficients of the diagonals of the ParPs--see this MO-Q. However, from Fig. 25 it's evident that the facets can be separated into at least two distinct geometric shapes. For the distinct types of facets of the associahedra (squares and pentagons) and the permutahedra (squares and hexagons) such a phenomenon is reflected in distinct monomials of the ParPs associated with these facets--the ParPs for compositional inversion of o.g.f.s and multiplicative inversion of e.g.f.s, respectively. In fact (in response to Sam's comment), the ParPs associated with these two families of convex polytopes encode in their monomials ALL of the geometric / topological aspects of the faces (1-dim vertices, 2-dim edges, 3-dim polygons, etc.). This close linkage between the combinatorics of the algebra / partition polynomials and the geometry / topology of the polytopes suggest looking for a parallel in the cluster complexes, so

Does anyone have illustrations of the lower order cluster complexes that clearly depict the distinct families of facets (as well as lower dimensional faces)?

(I know how to analytically refine the number of facets a couple of ways that have combinatorial significance.)