Bounds on the Number of Numerical Semigroups of a Given Genus

TL;DR

This paper establishes new upper and lower bounds on the number of numerical semigroups for each genus using combinatorics on multisets, significantly improving previous bounds.

Contribution

It provides the first bounds relating the count of numerical semigroups to Fibonacci numbers, refining the understanding of their growth.

Findings

Lower bound: n_g ≥ 2F_g

Upper bound: n_g ≤ 1 + 3·2^{g-3}

Bounds are tight and improve previous estimates.

Abstract

Combinatorics on multisets is used to deduce new upper and lower bounds on the number of numerical semigroups of each given genus, significantly improving existing ones. In particular, it is proved that the number $n_g$ of numerical semigroups of genus $g$ satisfies $2F_{g}\leq n_g\leq 1+3\cdot 2^{g-3}$, where $F_g$ denotes the $g$th Fibonacci number.