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.
Highlights
A numerical semigroup is a subset S ⊂ Z 0 that is closed under addition
Part (a) of Theorem 1 follows from standard arguments in probabilistic combinatorics (Theorem 5), parts (b) and (c) follow, surprisingly, from the construction of a very natural shellable simplicial complex (Definition 8) whose facets are in bijection with irreducible numerical semigroups of a fixed Frobenius number (Definition 7)
Before proving the remaining parts of Theorem 1, we introduce in Definition 8 a simplicial complex whose combinatorial properties govern several questions arising from the ER-type model for sampling random numerical semigroups
Summary
A numerical semigroup is a subset S ⊂ Z 0 that is closed under addition (we do not require S to have finite complement in Z 0). Part (a) of Theorem 1 follows from standard arguments in probabilistic combinatorics (Theorem 5), parts (b) and (c) follow, surprisingly, from the construction of a very natural shellable simplicial complex (Definition 8) whose facets are in bijection with irreducible numerical semigroups of a fixed Frobenius number (Definition 7). As it turns out, some of the probabilities involved in determining the expected values above require precisely the h-vector (in the sense of algebraic combinatorics [21]) for this simplicial complex. Through the h-vector, we distinguish parts (b) and (c) of Theorem 1 (Corollary 26) and estimate the finite expectations (Theorem 30)
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