From "[The multiple facets of the associahedra](http://www.claymath.org/library/academy/LectureNotes05/Lodaypaper.pdf)" by Loday:



> 
> Let us consider the formal power series
> 
> 
> $$f(x) = x+a\_1 x^2 +a\_2 x^3 + \cdots+ a\_n x^{n+1} + \cdots$$
> 
> 
> and let
> 
> 
> $$ g(x) = x+b\_1 x^2 + b\_2 x^3 + \cdots + b\_n x^{n+1} + \cdots$$
> 
> 
> be its inverse for composition, that is,
> 
> 
> $$f(g(x)) = x.$$
> 
> 
> The coefficient $b\_n$ is a polynomial in $a\_1$ to $a\_n$. In low dimension, one gets
> 
> 
> $$b\_1 = a\_1$$
> $$b\_2 = 2a\_1^2 - a\_2$$
> $$b\_3 = -5a\_1^3+5a\_1 a\_2-a\_3$$
> $$b\_4 = 14a\_1^4 -21 a\_1^2 a\_2 +6a\_1 a\_3 +3a\_2^2 -a\_4$$
> 
> 
> and more generally
> 
> 
> $$b\_n = \sum (-1)^{\sum n\_i} \lambda(n\_1,...,n\_k) a\_1^{n\_1} \cdots a\_n^{n\_k},$$
> 
> 
> where the sum is extended to all the k-tuples of integers $(n\_1,...,n\_k)$ so that $n\_1 +2n\_2 + \cdots +k n\_k =n$. Here the coefficient $\lambda(n\_1,...,n\_k)$ is the number of cells of the associahedron $K^{n-1} $ that are isomorphic to the cartesian product $(K^0)^{n\_1} \times \cdots \times(K^{k-1})^{n\_k}$.
> 
> 
> 


In other words, for example, $b\_4$ is related to a refined f-vector (face-vector) for the 3-D Stasheff polytope, or [3-D associahedron](https://en.m.wikipedia.org/wiki/Associahedron), with **14** vertices (0-D faces), **21** edges (1-D faces), **6** pentagons (2-D faces), **3** rectangles (2-D faces), **1** 3-D polytope (3-D faces). Subtracting one from the index of $a\_n$, and ignoring the resulting indeterminates with indices with values less than one, allows one to read off the geometry of the associahedron from cartesian products of the lower dimensional associahedra, e.g., $3\: a^2\_2$ becomes $3\: a^2\_1$, the cartesian product of the 1-D associahedron with itself, which is a tetragon, or square in some reps.


Loday further asserts:



> 
> There exists a short operadic proof of the above formula which explicitly involves the parenthesizings, but it would be interesting to find one which involves the topological structure of the associahedron.
> 
> 
> 


**QUESTION: What references contain proofs of the above argument relating the monomials of the inversion formula to distinct faces of an associahedron as cartesian products of the lower dimensional associahedra?**


Similar relationships hold, with a shift in indices, for the [permutahedra](https://en.m.wikipedia.org/wiki/Permutohedron) (see this [MSE-Q](https://math.stackexchange.com/questions/1785526/bijective-mapping-between-face-polytopes-of-permutohedra-and-partitions-of-integ)) and [noncrossing partitions](https://en.m.wikipedia.org/wiki/Noncrossing_partition) (see OEIS [A134264](https://oeis.org/A134264)).



Loday's characterization and enumeration of faces of associahedra (Stasheff polytopes)
I believe that the proof to which Loday is referring is the one that appears for Proposition 13.11.7 in [*Algebraic Operads*](https://books.google.com/books?id=s9Usr1BK4pwC&pg=PA550&lpg=PA550&dq=associahedron+inverse+power+series&source=bl&ots=uKg6O09Kb5&sig=Ih2h3Bd4qNXURDcfVSkGg0COrY4&hl=en&sa=X&ved=0ahUKEwjtyIz967vVAhVh04MKHXMOAhgQ6AEIXDAI#v=onepage&q=associahedron%20inverse%20power%20series&f=false). In this book, the coefficients $\lambda(n\_1,\dots,n\_k)$ are described as the number of planar rooted trees with certain properties. Such trees are in bijection with the Cartesian products of interest. Also, you might find the example following Corollary 9.2 of [this paper](https://arxiv.org/pdf/math/0411542.pdf) helpful.


