Counting numerical semigroups by genus and even gaps

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 $N_γ(g)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $γ$. We show that $N_γ(g)=N_γ(3γ)$ for $γ\leq \lfloor g/3\rfloor$ and $N_γ(g)=0$ for $γ> \lfloor 2g/3\rfloor$; thus the question above is true provided that $N_γ(g+1) > N_γ(g)$ for $γ= \lfloor g/3 \rfloor +1, \ldots, \lfloor 2g/3\rfloor$. We also show that $N_γ(3γ)$ coincides with $f_γ$, the number introduced by Bras-Amorós in conection with semigroup-closed sets. Finally, the stronger possibility $f_γ\sim φ^{2γ}$ arises being $φ= (1+\sqrt{5})/2$ the golden number.