The associahedra are wonderful things discovered by Jim Stasheff around 1963 but even earlier by Dov Tamari in his thesis. They hold the keys to understanding ‘associativity up to coherent homotopy’ in exquisite combinatorial detail.

But do they still hold more secrets? I think so!

Of course, what’s a secret to me may be well-known and boring to you. But this result by Marcelo Aguiar and Federico Ardilla intrigues and excites me!

Take a formal power series like this:

C ( x ) = x + c 1 x 2 + c 2 x 3 + ⋯ C(x) = x + c_1 x^2 + c_2 x^3 + \cdots

If you take its right inverse under composition, meaning the power series D D with

$$ C(D(x)) = x you get another formal power series of the same type : you get another formal power series of the same type: D ( x ) = x + d 1 x 2 + d 2 x 3 + ⋯ D(x) = x + d_1 x^2 + d_2 x^3 + \cdots How are the How are the dn related to the related to the cn ? Do some calculations : ? Do some calculations: d 1 = − c 1 d_1 = - c_1 d 2 = − c 2 + 2 c 1 2 d_2 = -c_2 + 2 c_1^2 d 3 = − c 3 + 5 c 2 c 1 − 5 c 1 3 d_3 = -c_3 + 5 c_2 c_1 - 5 c_1^3 d 4 = − c 4 + 6 c 3 c 1 + 3 c 2 2 − 21 c 2 c 1 2 + 14 c 1 4 d_4 = -c_4 + 6 c_3 c_1 + 3 c_2^2 - 21 c_2 c_1^2 + 14 c_1^4 What are these coefficients ? They ′ re controlled by the associahedra ! For example , the 3 − dimensional associahedron looks like this : Unknown character a href = Unknown character https : / / www . discretization . de / en / projects / A 11 / Unknown character Unknown character Unknown character img width = Unknown character 500 Unknown character src = Unknown character http : / / math . ucr . edu / home / baez / mathematical / associahedron . png Unknown character alt = Unknown character Unknown character / Unknown character Unknown character / a Unknown character It has : • 1 three − dimensional face , • 6 pentagonal and 3 square faces , • 21 edges , and • 14 vertices . So , if we call the What are these coefficients? They're controlled by the associahedra! For example, the 3-dimensional associahedron looks like this: <a href = "https://www.discretization.de/en/projects/A11/"> <img width = "500" src = "http://math.ucr.edu/home/baez/mathematical/associahedron.png" alt = ""/></a> It has: • 1 three-dimensional face, • 6 pentagonal and 3 square faces, • 21 edges, and • 14 vertices. So, if we call the n − dimensional associahedron -dimensional associahedron {\mathbf{c}}{n-1} , then , then \mathbf{c}1 is a point , is a point, \mathbf{c}2 is an interval , is an interval, \mathbf{c}3 is a pentagon , and the 3 d associahedron is a pentagon, and the 3d associahedron {\mathbf{c}}4 has • 1 face shaped like has • 1 face shaped like {\mathbf{c}}4 , • 6 faces shaped like , • 6 faces shaped like {\mathbf{c}}3 \times \mathbf{c}1 and 3 faces shaped like and 3 faces shaped like {\mathbf{c}}2 \times {\mathbf{c}}2 , • 21 faces shaped like , • 21 faces shaped like {\mathbf{c}}2 \times \mathbf{c}1 \times \mathbf{c}1 , and • 14 faces shaped like , and • 14 faces shaped like \mathbf{c}1 \times \mathbf{c}1 \times \mathbf{c}1 \times \mathbf{c}_1 . All this face information is packed into the formula we saw : . All this face information is packed into the formula we saw: d 4 = − c 4 + 6 c 3 c 1 + 3 c 2 2 − 21 c 2 c 1 2 + 14 c 1 4 d_4 = -c_4 + 6 c_3 c_1 + 3 c_2^2 - 21 c_2 c_1^2 + 14 c_1^4 $

Why is this happening, and what can we do with it? I don’t know! I should start by reading this:

Marcelo Aguiar, Federico Ardila, Hopf monoids and generalized permutahedra.

They have very similar results for permutahedra and other delicious polytopes. The permutahedra show up when you invert formal power series with respect to multiplication, rather than composition!