Fibonacci-like growth of numerical semigroups of a given genus
By Alex Zhai. 2011
84 citations recorded by the source corpus.
AbstractWe give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if ng 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 ng. 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.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/37ab178ff52f921add11b40a5d5383113b50bebb
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In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
In Geometry and Topology
Citations in the corpus
- Wilf’s conjecture and Macaulay’s theorem (2017, background): “The importance of the former case stems from a recent result of Zhai stating that, asymptotically as the genus g(S) goes to infinity, the proportion of numerical semigroups S satisfying c(S) ≤ 3m(S) tends to 1 [24].”