Random numerical semigroups and a simplicial complex of irreducible semigroups (Q1627215)

Summary: We examine properties of random numerical semigroups under a probabilistic model inspired by the Erdős-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.