On the kappa ring of $\overline{M}_{g,n}$

Let $\kappa_e(\overline{M}_{g,n})$ denote the kappa ring of $\overline{M}_{g,n}$ in codimension $e$. For $g,e\geq 0$ fixed, as the number $n$ of the markings grows large we show that the rank of $\kappa_e(\overline{M}_{g,n})$ is asymptotic to $$\frac{{n+e\choose e}{g+e\choose e}}{(e+1)!}\simeq \frac{{g+e\choose e}n^e}{e!(e+1)!}.$$ When $g\leq 2$ we show that a kappa class $\kappa$ is trivial if and only if the integral of $\kappa$ against all boundary strata is trivial. For $g=1$ we further show that the rank of $\kappa_{n-d}(\overline{M}_{1,n})$ is equal to $|P_1(d,n-d)|$, where $P_i(d,k)$ denotes the set of partitions $p=(p_1,...,p_\ell)$ of $d$ such that at most $k$ of the numbers $p_1,...,p_\ell$ are greater than $i$.