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 [A126216] = [A001263][A007318]= Narayana * Pascal $= [N][P]$ as 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.
12) The shifted reciprocal of the o.g.f. of the refined Euler characteristic partition polynomials of the associahedra give the formal free cumulants of free probability theory, and the polynomials are proportional to a partial derivative of the free cumulants (added 1/20/22)
See the MO-Q "Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory".
13) The associahedra and noncrossing partitions are dual geometric and analytic constructs (Added 7/19/2022)
The signed generalization/refinement of item 9 above is $[A] = [N][R] =$ [A133437] = [A111785] = [A134264][signed A263633], where, e.g., $[R][a]$ represents substitution of the infinite set of indeterminates $(a_1,a_2,…)$ for those of the reciprocal partition polynomials (for o.g.f.s) defined by $x/f(x) = 1/(1+c_1x+c_2x^2 + \cdots) = R_0 + R_1(c_1) x + R_2(c_1,c_2) x^2 + \cdots$; the set of partition polynomials of $[A]$ are the refined Euler characteristic polynomials for the associahedra; and $[N]$, the refined Narayana, or noncrossing partition, polynomials. (The indeterminates and partition polynomials of the associahedra polynomials A1334347 must be re-indexed, i.e., shifted by -1, with (1') = 1.) Note $[R]^2 =[I]=[A]^2$ is the identity transformation under indeterminate substitution/composition, so also $[A][R] = [N]$.
In addition, $[I]=[A][R][R][A] = [N][R][A] = [R][A][A][R] = [R][A][N] $ implies $[R][A] = [N]^{-1}$, the inverse of $[N]$. The pair of inverses define the free moments and cumulants of free probability theory in terms of each other, so we can connect the associahedra to free probability as well along with its connections to random matrix theory and quantum fields.
The e.g.f. equivalent is $[L] = [E][P]$, where $[L]$ is the set of classic Lagrange inversion polynomials A134685, associated with weighted phylogenetic trees; $[E]$, the refined Eulerian polynomials A145271; and $[P]$, the refined Euler characteristic polynomials $P_n(d_1,…,d_n)$ A133314 of the permutahedra, giving the Taylor series coefficients of the reciprocal $1/h(x)$ of the e.g.f. $h(x) = 1 + d_1 x +d_2 \frac{x^2}{2!}+\cdots.$ We have a dual set of generators $[E]$ and $[E]^{-1}$ for an infinite group as well since $[L]^2 = [I] = [P]^2$.
The two sets of compositional inversion polynomials $[L]$ and $[A]$ are related by a simple scaling of the indeterminates by the factorials just as formal e.g.f.s are related to formal o.g.f.s–a simple Borel-Laplace transform term by term–and so are the sets of multiplicative inversion polynomials $[R]$ and $[P]$ .
From the four sets of partition polynomials $[A]$,$[L]$,$[N]$, and $[E]$ issue compositional inverses of series while from $[R]$ and $[P]$, multiplicative inverses. This equivalence is the integrating thread of a tapestry of geometric/topological and algebraic/analytic constructs from polytopes and trees to moduli spaces, punctured Riemann surfaces, and characteristic classes to quadratic operads and Lie derivatives/infinitesimal generators (and, naturally, up on the tapestry is quantum physics, e.g., see this MO-Q).
(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, … .
Best Answer
If $Q$ is the quiver $A_n$ with linear orientation, then the Stasheff associahedron appears as the polytope with vertices the basic tilting $kQ$-modules and faces the faithful basic exceptional $kQ$-modules. The partial order on faces is defined by $M \leq N$ whenever $N$ is a a direct summand of $M$. This gives a poset isomorphic to the one in your first example.
References:
Buan, Aslak Bakke; Krause, Henning. Tilting and cotilting for quivers of type $\tilde A_n$, section 3 and appendix A.
Buan, Aslak Bakke; Marsh, Robert; Reineke, Markus; Reiten, Idun; Todorov, Gordana. Tilting theory and cluster combinatorics, section 4.