Counting numerical sets with no small atoms

TL;DR

This paper investigates the asymptotic behavior of the proportion of numerical sets with a specific atom monoid structure, revealing convergence to approximately 48.44% and analyzing related generating functions.

Contribution

It introduces the sequence $\u03b3_g$ for counting numerical sets with a given atom monoid and proves its convergence, providing new insights into the structure of numerical sets.

Findings

The sequence $3g$ decreases and converges to about 0.4844.

The generating function for $3g$ has specific singularities.

Parallel results for symmetric numerical sets are obtained.

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$ for all $s \in S$}$. Let $\gamma_g$ be the number of numerical sets $S$ having $A(S) = \set{0} \cup (g,\infty)$ divided by the total number of numerical sets with Frobenius number $g$. We show that the sequence $\set{\gamma_g}$ 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 $\set{\gamma_g}$. Parallel results are obtained for the ratio $\gsymm{g}$ 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.