On Numerical Semigroups with Almost-Maximal Genus

Abstract

A numerical semigroup is a cofinite subset of N₀, containing 0, that is closed under addition.  Its genus is the number of nonnegative integers that are missing.  A numerical set is a similar object, not necessarily closed under addition.  If T is a numerical set, then A(T)={n in N₀ : n+T is a subset of T} is a numerical semigroup.  Recently a paper appeared counting the number of numerical sets T where A(T) is a numerical semigroup of maximal genus.  We count the number of numerical sets T where A(T) is a numerical semigroup of almost-maximal genus, i.e. genus one smaller than maximal.