Elasticity in Apéry Sets

Abstract

A numerical semigroup S is an additive subsemigroup of the nonnegative integers, containing zero, with finite complement. Its multiplicity m is its smallest nonzero element. The Apéry set of S is the set of elements $\text{Ap}(S)=\{n\in S:n-m\notin S\}$. Fixing a numerical semigroup, we ask how many elements of its Apéry set have nonunique factorization and define several new invariants.

• MSC: Primary 20M14
• Secondary 05A16
• 11P99

Acknowledgments

The authors would like to acknowledge the support of NSF-REU grant 1851542. The authors would also like to thank Christopher Preuss for his help with Sage code for visualizing Apéry posets, Nathan Kaplan for many helpful conversations, and the anonymous referees for their useful suggestions.