Monotonicity of the number of numerical semigroups by genus

Prove that the number n_g of numerical semigroups of genus g satisfies n_g≤n_{g+1} for every nonnegative integer g.

Background

The total number n_g of numerical semigroups of genus g is discussed as a fundamental enumeration problem. The paper identifies the nondecreasing-genus-count conjecture from the cited literature as a motivating unresolved question.

References

Many of these publications are motivated for attempting to respond to the conjecture established in , which argues that $n_g\leq n_{g+1}$.

— Internal numerical semigroups  (2608.19984 - Casas et al., 20 Aug 2026) in Section 1, Introduction

If $k\in\mathbb{N}$, then $#{\mathscr{I}(\text{gen}=k)} \leq #{\mathscr{I}(\text{gen}=k+1)}$.

— Internal numerical semigroups  (2608.19984 - Casas et al., 20 Aug 2026) in Conjecture 6, Section 2