Complementary Numerical Sets

Abstract

A numerical set $S$ is a cofinite subset of $\mathbb{N}$ which contains $0$. We use the natural bijection between numerical sets and Young diagrams to define a numerical set $\tilde{S}$, such that their Young diagrams are complements. We determine various properties of $\tilde{S}$, particularly with an eye to closure under addition (for both $S$ and $\tilde{S}$), which promotes a numerical set to become a numerical semigroup.

12 pages