Figure 25 on pg. 101 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. for $m$ a negative integer in my terminology and positive in theirs.

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--squares and pentagons--of the 3-dimensional associahedron, a convex polytope, and the facets--squares and hexagons--of the 3-D permutahedron, another convex polytope, 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 polytopes encode in their monomials ALL of the geometric / topological aspects of the faces (1-D vertices, 2-D edges, 3-D polygons, etc.).

This phenomenon extends in a more complicated fashion to the geometric combinatorics of non-polytope constructs, such as the complexes associated with the tropical Grassmannians $G(2,n)$ and the algebraic combinatorics of the classic Lagrange ParPs for compositional inversion of e.g.f.s as well as to the 'sprigs' of Ardila and the refined Pascal ParPs for multiplicative inversion of o.g.f.s.

This close linkage between the algebraic combinatorics of ParPs and the combinatorics of related geometric constructs, whether polytopes or more complicated complexes, suggest exploring parallels for the cluster complexes and associated ParPs or modifications of both or either of the two. I'm not familiar with CCs, so to suggest and test conjectural modifications, I'm hoping someone can answer the following question;

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

(I know two or three ways to analytically refine the Fuss-Narayana numbers, the coefficients of the diagonals of the $(m)$-Narayana ParPs, that enumerate the facets. Whether these can cleanly be or should be incorporated into the algebraic combinatorics is another matter.)