Inversion of series and the cohomology of the moduli spaces ${\mathcal{M}}_{0,n}^{\delta }$

For $n\ge 3$, let ${\mathcal{M}}_{0,n}$ denote the moduli space of genus 0 curves with $n$ marked points, and ${\bar{\mathcal{M}}}_{0,n}$ its smooth compactification. A theorem due to Ginzburg, Kapranov and Getzler states that the inverse of the exponential generating series for the Poincar'e polynomial of $H^{\bullet }({\mathcal{M}}_{0,n})$ is given by the corresponding series for $H^{\bullet }({\bar{\mathcal{M}}}_{0,n})$. In this paper, we prove that the inverse of the ordinary generating series for the Poincar'e polynomial of $H^{\bullet }({\mathcal{M}}_{0,n})$ is given by the corresponding series for $H^{\bullet }({\mathcal{M}}_{0,n}^{\delta })$, where $$\mathcal{M}${0,n}\subset $\mathcal{M}$^{\delta}{0,n} \subset \overline{$\mathcal{M}$}_{0,n}$ is a certain smooth affine scheme.