Counting numerical sets with no small atoms

Abstract

A numerical set $S$ with Frobenius number $g$ is a set of integers with $\min (S)=0$ and $\max(\Zbb - S)=g$, and its atom monoid is $A(S) = \setpres{n \in \Zbb}{$n+s \in S$forall$s \in S$}$. Let ${\gamma }_g$ be the number of numerical sets $S$ having $A(S)=\{0\}\cup (g,\infty )$ 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 ${\gamma }_{\infty }\approx .4844$ (with accuracy to within $.0050$). We also examine the singularities of the generating function for $\{{\gamma }_g\}$. Parallel results are obtained for the ratio $\gsymm{g}$ of the number of symmetric numerical sets $S$ with $A(S)=\{0\}\cup (g,\infty )$ 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.

19 pages, 5 figures