On the Betti numbers, Poincar\'e polynomials, and Euler characteristics of $\overline{\mathcal M}_{0,n}$

In this paper, we revisit the Poincaré polynomials, Betti numbers, and Euler characteristics of the Deligne-Mumford moduli spaces $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves. We give elementary derivations of two recent closed formulas for their Poincaré polynomials, due respectively to Aluffi-Marcolli-Nascimento (arXiv:2406.13095) and to Eur-Ferroni-Matherne-Pagaria-Vecchi (arXiv:2504.16776). Our approach shows that both formulas are already implicit in the generating-series results of Getzler and Manin, and can be extracted from them by elementary manipulations of generating functions, the binomial series, and standard identities for Stirling numbers. Beyond these new derivations, the same method also yields new linear recurrence relations for refined invariants associated with these Poincaré polynomials, namely distinguished summands and a bivariate refinement. As a further consequence, we obtain two additional formulas for the Betti numbers, not previously recorded in this form.
We also study the Euler characteristics $\chi(\overline{\mathcal M}_{0,n})$. Using the Taylor expansion of a suitable branch of the Lambert $W$-function, we show that their sequence is obtained by evaluating complete Bell polynomials at an explicit auxiliary integer sequence. This Bell-polynomial representation yields Hessenberg determinantal formulas and a new linear recursion, distinct from the well-known quadratic Keel-Manin recursion. It also provides an explicit extraction of the Euler characteristics from the Lambert $W$-function expression considered by Aluffi-Marcolli-Nascimento. Finally, we refine the Manin-Zagier asymptotic estimate for these Euler characteristics by computing the full asymptotic expansion.