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 $N_{\gamma }(g)$ 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 $\gamma \le \lfloor g/3\rfloor$ 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 $\gamma =\lfloor g/3\rfloor +1,\dots ,\lfloor 2g/3\rfloor$. We also show that $N_{\gamma }(3\gamma )$ coincides with $f_{\gamma }$, the number introduced by Bras-Amorós in conection with semigroup-closed sets. Finally, the stronger possibility $f_{\gamma }\sim {\phi }^{2\gamma }$ arises being $\phi =(1+\sqrt{5})/2$ the golden number.

17 pages