Fibonacci-like growth of numerical semigroups of a given genus

Abstract

We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if n _g is the number of numerical semigroups of genus g, we prove that

$$\lim_{g \rightarrow \infty} n_g \varphi^{-g} = S $$

where $\varphi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio and S is a constant, resolving several related conjectures concerning the growth of n _g . In addition, we show that the proportion of numerical semigroups of genus g satisfying f<3m approaches 1 as g→∞, where m is the multiplicity and f is the Frobenius number.

Appendix

We fill in here some of the computational details of the proofs of Lemmas 18 and 19. The matrix calculations of this section were done using Sage.⁵ Recall that in the proof of Lemma 18 we made the definition

$$W_n = \left[ \begin{array}{c@{\quad}c@{\quad}c} 1 & 1 & 1 \\ \end{array} \right] \left[ \begin{array}{c@{\quad}c@{\quad}c} 1 & 1 & 1 \\ \varphi^{-1} & 1 & 1 \\ \varphi^{-2} & 1 & \varphi^{-1} \\ \end{array} \right]^{n - 1} \left[ \begin{array}{c} 1 \\ 1 \\ 1 \\ \end{array} \right]. $$

We now justify in detail the claim that W _n ≤3×1.618²ⁿ⁻². Note that by explicit computation,

By the Cayley-Hamilton theorem, we obtain the recurrence

$$W_{n + 3} = (2.618 \cdots) W_{n + 2} - (0.236 \cdots) W_{n + 1} - (0.146 \cdots) W_n $$

for n≥1. Thus, W _n+1≤2.619W _n <1.618² W _n for each n≥3, and by induction, W _n ≤3×1.618²ⁿ⁻², as desired.

A similar claim was made in the proof of Lemma 19. In that proof, we defined

$$W_n = \left[ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array} \right] \left[ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & \varphi^{-1} & 1 & \varphi^{-1} & 1 & \varphi^{-2} \\ 1 & 1 & 1 & 1 & 1 & 1 \\ \varphi^{-1} & \varphi^{-2} & 1 & \varphi^{-1} & 1 & \varphi^{-3} \\ \varphi^{-1} & \varphi^{-1} & 1 & 1 & 1 & \varphi^{-1} \\ \varphi^{-1} & \varphi^{-2} & \varphi & 1 & \varphi & \varphi^{-3} \\ \varphi & \varphi & \varphi & \varphi & \varphi & \varphi \\ \end{array} \right]^{n - 1} \left[ \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \\ \varphi \\ \varphi \\ \end{array} \right]. $$

It was claimed that W _n ≤(4+2φ)×1.618⁴ⁿ⁻⁴. As before, we first verify this for small values of n.

We will prove by induction that W _n+1≤6.8W _n for all n≥1. This can be seen by direct verification for n≤4. Proceeding inductively, for n>4, we have by the Cayley-Hamilton theorem,

where we have used implicitly the basic inequalities W _k ≥0 and W _k+1≥W _k for all k≥1. Noting that 1.618⁴>6.8, it is now immediate that W _n ≤(4+2φ)×1.618⁴ⁿ⁻⁴ for all n≥1.