Counting numerical semigroups by Frobenius number, multiplicity, and depth

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 $q\ge 3$, we show that, given $(q-1)m<f<qm$, the number of numerical semigroups with Frobenius number $f$ and multiplicity $m$ is\[\left(\left\lfloor \frac{(q+2)^2}{4} \right\rfloor^{α/2} \left \lfloor \frac{(q+1)^2}{4} \right\rfloor^{(1-α)/2}\right)^{m + o(m)}\] 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.

23 pages, 6 figures; accepted to Comb. Theory, incorporated referee comments

References in corpus (2)

• Fibonacci-Like Behavior of the Number of Numerical Semigroups of a Given Genus
• Sub-Fibonacci behavior in numerical semigroup enumeration