Calculations of integral homology of $\mathcal{M}_{g, n}$ occur in Abhau, Bodigheimer, Ehrenfried (p. 3) or Godin (p. 4). Of course, in the stable range (degrees $3* \leq 2g$) the rational cohomology is entirely known (by Mumford's conjecture, now theorem), and is a polynomial algebra on a single generator in each even degree. Thus the Betti numbers in this range are given by partition functions.

I should mention that Godin's results were obtained using the complex of fat graphs, which is probably equal to Kontsevich's graph complex for the associative operad.

Calculations of integral homology of $\mathcal{M}_{g, n}$ occur in Abhau, Bodigheimer, Ehrenfried (p. 3) or Godin (p. 4). Of course, in the stable range (degrees $3* \leq 2g$) the rational cohomology is entirely known (by Mumford's conjecture, now theorem), and is a polynomial algebra on a single generator in each even degree. Thus the Betti numbers in this range are given by partition functions.

I should mention that Godin's results were obtained using the complex of fat graphs, which is probably equal to Kontsevich's graph complex for the associative operad.