Gapsets and numerical semigroups (Q2010627)

A \textit{numerical semigroup} \(S\) is a cofinite submonoid of \((\mathbb{N},+)\). The \textit{genus} of \(S\) is the cardinality \(|\mathbb{N}\setminus S|\). This paper is motivated by a long-standing conjecture of [\textit{M. Bras-Amorós}, Semigroup Forum 76, No. 2, 379--384 (2008; Zbl 1142.20039)] concerning the number \(n_g\) of numerical semigroups of genus \(g\): the Fibonacci-like inequality \[ n_{g-1}+n_{g-2}\leq n_g.\] The authors concentrate on a subset of the set of numerical semigroups of genus \(g\). The \textit{Frobenius number} of \(S\) is the number \(F(S) =\max(\mathbb{N}\setminus S)\), while the \textit{multiplicity} of \(S\) is the number \(m(S) = \min(S\setminus \{0\})\). Denote by \(n'_g\) the number of numerical semigroups of genus \(g\) such that \(F(S) <3m(S)\). The main result of the paper states that \[ n'_{g-1}+n'_{g-2} \leq n'_g \leq n'_{g-1}+n'_{g-2}+n'_{g-3}. \] This result provides evidence in favor of the conjecture above, since it is proved in [\textit{Y. Zhao}, Semigroup Forum 80, No. 2, 242--254 (2010; Zbl 1204.20080)] that \(\lim_{g \rightarrow \infty} \frac{n'_g}{n_g} = 1\). In order to prove the inequalities about semigroups satisfying \(F(S) <3m(S)\), the authors employ combinatorial constructions based on the notion of \textit{gapset filtration}, which is, roughly speaking, a way to encode the semigroup \(S\) by partitioning the set \(\mathbb{N}\setminus S\) according to the three intervals \( [1,m(S)-1]\), \( [m(S)+1,2m(S)-1]\), and \( [2m(S)+1,3m(S)-1]\).