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 for $m$ a negative integer in my nomenclature 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 (if I interpret them correctly). Similar phenomena are associated with other closely related sets of ParPs as noted in the background section below.
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?
Background
Similar phenomena occur for the distinct types of facets--squares and pentagons--of the 3-dimensional associahedron, a convex polytope (see. e.g., this MO-Q), and the facets--squares and hexagons--of the 3-D permutahedron, another convex polytope. This bifurcation is reflected in the two distinct monomials of the ParPs associated with these facets, which are 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.); i.e., there is a clean bijection between the monomials and the distinct geometric constructs.
In a less clean fashion, this phenomenon extends partially to the more complicated simplicial complexes the tropical Grassmannians $G(2,n)$ associated with the mono-variate Ward polynomials (A134991), a natural reduction of the set $[L]$ of classic Lagrange Parps (A134685) for compositional inversion of e.g.f.s. For details, see "Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials" by Price and Sokal. In seeking a potential generalization to the ParpS, scanning "The Tropical Grassmannian" by Speyer and Sturmfels, you can see a correspondence between the coefficients of $L_3 = -15 u_1^3 + 10 u_1 u_2 -u_3$ and the vertices and edges of the simplicial complex $T_5$, the Petersen graph, and a correspondence between a breakdown of $T_6$ in Example 4.1 on pg. 8 of S & S and $L_4 = 105 u_1^4 - 105 u_1^2 u_2 + 15 u_1 u_3 + 10 u_2^2 - u_4 $. A general bijection with the ParPs, however, is still an open problem, as Price and Sokal note.
This close linkage between the algebraic / analytic combinatorics of ParPs and the combinatorics of related geometric constructs, whether polytopes or more complicated complexes, suggest exploring parallels for the Cataland cluster complexes and associated ParPs. I'm not familiar with CCs, so to suggest and test conjectural modifications, I've posed the request for illustrations and certainly further references would also be appreciated.
The model presented in Cataland is different from other models associated with the $(m)$-Narayana ParPs. For $m>1$, analytic / algebraic / geometric combinatorics of the refined $(m)$-Narayana ParPs are inherited from those of the refined Narayana ParPs (see. e.g., this MO-Q). The same applies for $m < 1$ with respect to the inverse refined Narayana polynomials except this set doesn't have quite the variety of geometric models as the set of refined Narayana ParPS does (see this MO-Q).
(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 Cataland facets. Whether these can cleanly be or should be incorporated into the more encompassing algebraic combinatorics is another matter.)
Edit April 18, 2023:
Nathan Williams responded to an email I sent him with an alternate dual representation of his cluster complexes that seems to correlate with at least a variant of the reduced polynomials for the $(m)-$Narayana partition polynomials as face polynomials. I'll encourage him to present his observations as an answer here. This correlate also with Armstrong's thesis "Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups".
