Numerical semigroup tree: type-representation

TL;DR

This paper introduces a new representation of the numerical semigroup tree based on type, revealing constant counts of semigroups for certain genus and type values, and explores properties like unimodality and leaf behavior through computational experiments.

Contribution

It presents a novel depiction of the numerical semigroup tree using type, and investigates the distribution and properties of semigroups with computational evidence.

Findings

Number of semigroups of genus g and type t is constant when t is close to g as g increases.

Unimodality of various sequences related to semigroup properties.

Behavior of leaves in the numerical semigroup tree.

Abstract

In this paper, we introduce a new depicting of the so-called numerical semigroup tree $\mathcal T$. By exploring computationally this improved picture, relying on the type notion of a semigroup, we found that the number of semigroups of genus $g$ and type $t$ is constante when $t$ is close to $g$ while $g$ grows. We also study the unimodality of various sequences as well as the behavior of the leaves in $\mathcal T$. We put forward several conjectures that are supported by various computational experiments.