Counting numerical sets with no small atoms

A numerical set $S$ with Frobenius number $g$ is a set of integers with $\min (S)=0$ and $\max (\text{Zbb}-S)=g$, and its atom monoid is $A(S) = \setpres{n \in \Zbb}{$n+s\in S$ for all $s\in S$}$\mathrm{.}Let$\gamma_g$ be the number of numerical sets $S$ having $A(S)=\{0\}\cup (g,\mathrm{\infty })$ divided by the total number of numerical sets with Frobenius number $g$. We show that the sequence $\{{\gamma }_g\}$ is decreasing and converges to a number $\gamma_\infty \approx .4844$(withaccuracytowithin$.0050$). We also examine the singularities of the generating function for $\{{\gamma }_g\}$. Parallel results are obtained for the ratio $\text{gsymm}g$ of the number of symmetric numerical sets $S$ with $A(S) = \set{0} \cup (g,\infty)$ by the number of symmetric numerical sets with Frobenius number $g$. These results yield information regarding the asymptotic behavior of the number of finite additive 2-bases.