Combinatorial Theory

In 2013, Zhai proved that most numerical semigroups of a given genus have depth at most \(3\) and that the number \(n_g\) of numerical semigroups of a genus \(g\) is asymptotic to \(S\varphi^g\), where \(S\) is some positive constant and \(\varphi \approx 1.61803\) is the golden ratio. In this paper, we prove exponential upper and lower bounds on the factors that cause \(n_g\) to deviate from a perfect exponential, including the number of semigroups with depth at least \(4\). Among other applications, these results imply the sharpest known asymptotic bounds on \(n_g\) and shed light on a conjecture by Bras-Amorós (2008) that \(n_g \geq n_{g-1} + n_{g-2}\). Our main tools are the use of Kunz coordinates, introduced by Kunz (1987), and a result by Zhao (2011) bounding weighted graph homomorphisms.

Mathematics Subject Classifications: 20M14, 05A15, 05A16

Keywords: Numerical semigroup, genus, Kunz coordinate, graph homomorphism