From "The multiple facets of the associahedra" 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, 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 (see this MSE-Q) and noncrossing partitions (see OEIS A134264).

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