Combinatorial Theory

In 1990, Backelin showed that the number of numerical semigroups with Frobenius number \(f\) approaches \(C_i \cdot 2^{f/2}\) for constants \(C_0\) and \(C_1\) depending on the parity of \(f\). In this paper, we generalize this result to semigroups of arbitrary depth by showing there are \(\lfloor (q+1)^2/4 \rfloor^{f/(2q-2)+o(f)}\) semigroups with Frobenius number \(f\) and depth \(q\). More generally, for fixed \(q \geq 3\), we show that, given \((q-1)m ‹ f ‹ qm\), the number of numerical semigroups with Frobenius number \(f\) and multiplicity \(m\) is \[\left(\left\lfloor \frac{(q+2)^2}{4} \right\rfloor^{\alpha/2} \left \lfloor \frac{(q+1)^2}{4} \right\rfloor^{(1-\alpha)/2}\right)^{m + o(m)}\] where \(\alpha = f/m - (q-1)\). Among other things, these results imply Backelin's result, strengthen bounds on \(C_i\), characterize the limiting distribution of multiplicity and genus with respect to Frobenius number, and resolve a recent conjecture of Singhal on the number of semigroups with fixed Frobenius number and maximal embedding dimension.

Mathematics Subject Classifications: 05A16, 20M14

Keywords: Numerical semigroups, Kunz coordinates, graph homomorphisms