Counting numerical semigroups by Frobenius number, multiplicity, and depth

TL;DR

This paper generalizes the asymptotic enumeration of numerical semigroups by Frobenius number to include parameters like depth and multiplicity, providing new bounds, characterizations, and resolving a recent conjecture.

Contribution

It extends Backelin's asymptotic results to semigroups of arbitrary depth and multiplicity, offering explicit formulas and resolving a recent conjecture.

Findings

Number of semigroups with Frobenius number f and depth q grows as /(2q-2)+o(f)

Explicit formulas for the count of semigroups with fixed Frobenius number and multiplicity

Characterization of the limiting distribution of multiplicity and genus

Abstract

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.