Algebraic Structures on Polytopes

2. Compositional Inversion. Consider power series C(x) = x+ ∑ n≥2 cn−1x , D(x) = x+ ∑ n≥2 dn−1x n that are compositional inverses; that is, C(D(x)) = x. The first few coefficients of D(x) = C(x)〈−1〉 are: d1 = −c1 d2 = −c2 + 2c1 d3 = −c3 + 5c2c1 − 5c1 d4 = −c4 + 6c3c1 + 3c2 − 21c2c1 + 14c1 ∗Professor of Mathematics, San Francisco State University. Simons Professor, Mathematical Sciences Research Institute. Profesor Adjunto, Universidad de Los Andes. federico@sfsu.edu. Supported by NSF CAREER grant DMS-0956178 and Combinatorics grants DMS-0801075, DMS-1440140, and DMS-1600609. It is natural to ask: What do these coefficients count? Two families of polytopes hold the answers.