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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 & B 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.

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.

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".

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 and positive in theirs.

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., their 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 & B 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.

The model presented in Cataland is different from other models associated with the $(m)$-Narayana ParPs. For $m>1$, the 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).

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.

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 & B 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.

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).