Counting numerical semigroups by genus and even gaps

TL;DR

This paper introduces a new approach to counting numerical semigroups by genus using even gaps, investigates their growth pattern, and connects these counts to Fibonacci-like sequences.

Contribution

It establishes formulas relating the number of semigroups with fixed even gaps to known sequences and explores their asymptotic behavior.

Findings

The number of semigroups with a given number of even gaps follows specific formulas.

The counts relate to the sequence f_gamma introduced by Bras-Amoros.

Asymptotic growth of these counts is conjectured to follow the golden ratio.

Abstract

Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true that $n_{g+1}>n_g$? is investigated. Let $N_\gamma(g)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $\gamma$. We show that $N_\gamma(g)=N_\gamma(3\gamma)$ for $\gamma \leq \lfloor g/3\rfloor$ and $N_\gamma(g)=0$ for $\gamma > \lfloor 2g/3\rfloor$; thus the question above is true provided that $N_\gamma(g+1) > N_\gamma(g)$ for $\gamma = \lfloor g/3 \rfloor +1, \ldots, \lfloor 2g/3\rfloor$. We also show that $N_\gamma(3\gamma)$ coincides with $f_\gamma$, the number introduced by Bras-Amor\'os in conection with semigroup-closed sets. Finally, the stronger possibility $f_\gamma \sim \varphi^{2\gamma}$ arises being $\varphi = (1+\sqrt{5})/2$ the golden number.