Random numerical semigroups and a simplicial complex of irreducible semigroups

TL;DR

This paper studies properties of random numerical semigroups using a probabilistic model, establishing thresholds and bounds for key invariants, and introduces a shellable simplicial complex linking irreducible semigroups to probabilistic properties.

Contribution

It introduces a novel probabilistic model for numerical semigroups and constructs a shellable simplicial complex connecting irreducible semigroups with probabilistic characteristics.

Findings

Threshold function for cofiniteness identified

Expected embedding dimension, genus, Frobenius number bounded

Shellable simplicial complex constructed linking semigroups and probabilistic properties

Abstract

We examine properties of random numerical semigroups under a probabilistic model inspired by the Erdos-Renyi model for random graphs. We provide a threshold function for cofiniteness, and bound the expected embedding dimension, genus, and Frobenius number of random semigroups. Our results follow, surprisingly, from the construction of a very natural shellable simplicial complex whose facets are in bijection with irreducible numerical semigroups of a fixed Frobenius number and whose $h$-vector determines the probability that a particular element lies in the semigroup.