Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the associahedra themselves occur in many guises, so in some sense they are a multi-dimensional, geometric extension of the Catalans, that also pop up in a surprising variety of places (for a quick intro, see Jon McCammond, "Noncrossing partitions in surprising places").

Is there a compilation for the associahedra, like Richard Stanley's for the Catalans?

If not, what guises are you aware of?

Examples are relations to

**1) Partial orderings of partial associations of a list**

Reflected in the face structure of the associahedra, as described in McCammond's article.

**2) Partition polynomials for Lagrange (compositional) inversion of functions or formal power series and so to iterated "Lie" derivatives**

The partition polynomials are isomorphic to the face structure of the associahedra, see "The multiple facets of the associahedron" by J. Loday, and can be generated by an iterated (Lie) derivative $[d/df(x)]^n$, where $f$ is expressed in the indeterminates of a power series, or o.g.f., (OEIS-A133437, OEIS-A145271).

**3) Classification of separation coordinates/variables for the Hamilton-Jacobi equations**

In "Separation coordinates, moduli spaces and Stasheff polytopes" by K. Schobel and P. Veselov (nice figures!), the combinatorics of the polytopes tessellating the real version of the Deligne-Mumford-Knudsen moduli space $\bar{M}_{0,n+2}(R)$ of stable curves of genus zero with $n + 2$ marked points are used to describe the topology and algebraic geometry of the space of separation coordinates on the spheres $S^n$ and to classify the different canonical forms of these coordinates, or separation variables, for the Hamilton-Jacobi equation.

**4) Diagonalization of convex polygons into non-overlapping convex sub-polygons**

See the beautifully illustrated book *Discrete and Computational Geometry* (pg. 74) by S. Devadoss and J. O'Rourke. Also "Polygonal dissections and reversion of series" by A. Schuetz and G. Whieldon.

**5) Secondary polytopes--convex hull of the area vectors of all triangulations of a convex polygon**

See Devadoss and O'Rourke, pg. 79.

**6) The Fulton-MacPherson compactification space of the configuration space of n particles colliding on an interval--truncated simplices**

See D and O, pg. 241. Nested tubings also.

**7) Deformation of bordered surfaces with marked points**

See "Deformations of bordered surfaces and convex polytopes" by S. Devadoss, T. Heath, and W. Vipismikul.

**8) Cluster A algebras and coordinates for scattering amplitudes**

See "Cluster polylogarithms for scattering amplitudes" by J. Golden, M. Paulos, M. Spradlin, and A. Vlolovich.

**9) Schroeder lattice paths (marked Dyck paths, OEIS A126216)**

Enumerated by f-vectors of the associahedra (Narayana * Pascal, lower triangular matrices).

**10) Solutions to the inviscid Hopf-Burgers equation**

See "Toric topology of the Stasheff polytopes" by V. Buchstaber and also MO-Q145555.

**11) Coinverse (antipode) for a Hopf algebra**

Analogous to the Faa di Bruno Hopf algebra, but represented in the indeterminates of a power series/ordinary generating function rather than those of a Taylor series/exponential generating function. Then the coproduct is related to Lah partition polynomials rather than Bell partition polynomials and the antipode, to Lagrange inversion/series reversion for o.g.f.s and therefore to associahedra rather than e.g.f.s. and Whitehouse simplicial complexes. For the usual e.g.f. formulation of the Faa di Bruno Hopf algebra, see *Quantum Field Theory II Quantum Electrodynamics* (pg. 136) by E. Zeidler or "Combinatorial Hopf algebras in quantum field theory I" by H. Figueroa and J. Gracia-Bondi.

(Each topic could be elaborated upon. The links in the OEIS refer to algebras I'm not comfortable with. Please feel free to do so in an answer.)

I know several examples, the third being the most recent I've come across, but there are others who frequent this site who can easily state more accurately and succinctly than I other connections of the associahedra to operads, spaces homotopically equivalent to loop spaces, trees, dendriform algebra, moduli spaces, ... .

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