Random Numerical Semigroups and a Simplicial Complex of Irreducible Semigroups

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.