We provide a programmable recursive algorithm for the ${\mathbb{S}}_n$-representations on the cohomology of the moduli spaces ${\overline{\mathcal{M}}}_{0,n}$ of $n$-pointed stable curves of genus 0. As an application, we find explicit inductive and asymptotic formulas for the invariant part $H^{\ast }({\overline{\mathcal{M}}}_{0,n}/{\mathbb{S}}_n)$ and prove that its Poincaré polynomial is asymptotically log-concave. Based on numerical computations with our algorithm, we further conjecture that the sequence $\{H^{2k}({\overline{\mathcal{M}}}_{0,n})\}$ of ${\mathbb{S}}_n$-modules is equivariantly log-concave.