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.

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