Counting numerical sets with no small atoms
| dc.creator | Marzuola, Jeremy | |
| dc.creator | Miller, Andy | |
| dc.date | 2008-05-22 | |
| dc.date.accessioned | 2026-07-07T09:40:25Z | |
| dc.date.available | 2026-07-07T09:40:25Z | |
| dc.description | 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 $γ_g$ 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 $\set{γ_g}$ is decreasing and converges to a number $γ_\infty \approx .4844$ (with accuracy to within $.0050$). We also examine the singularities of the generating function for $\set{γ_g}$. 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. | |
| dc.description | 19 pages, 5 figures | |
| dc.identifier | https://arxiv.org/abs/0805.3493 | |
| dc.identifier | http://arxiv.org/abs/0805.3493 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/161480 | |
| dc.subject | Combinatorics | |
| dc.subject | Number Theory | |
| dc.subject | 05A16; 05A15; 05A17 | |
| dc.title | Counting numerical sets with no small atoms | |
| dc.type | text |