Bounds on the Number of Numerical Semigroups of a Given Genus

Combinatorics on multisets is used to deduce new upper and lower bounds on the number of numerical semigroups of each given genus, significantly improving existing ones. In particular, it is proved that the number $n_g$ of numerical semigroups of genus $g$ satisfies $2F_{g}\leq n_g\leq 1+3\cdot 2^{g-3}$, where $F_g$ denotes the $g$th Fibonacci number.

💡 Research Summary

The paper addresses the long‑standing problem of estimating the number (n_g) of numerical semigroups of a given genus (g). A numerical semigroup is a subset (\Lambda\subseteq\mathbb N_0) containing 0, closed under addition, and with a finite complement; the size of the complement is the genus, while the largest gap is the Frobenius number (f). Existing literature provides the trivial bounds (n_0=n_1=1) and the Catalan bound (n_g\le C_g) (where (C_g) is the (g)‑th Catalan number). Moreover, empirical data suggest that (n_g) behaves like the Fibonacci sequence, leading to conjectures about asymptotic ratios.

The author introduces two families of recursively defined multisets, (A_g) and (B_g), which serve as combinatorial proxies for lower and upper estimates of (n_g).

Definition of (A_g).
Start with (A_2={1,3}). For (g>2) set
\