Almost-symmetric numerical semigroups A numerical semigroup is almost-symmetric () if its genus is the arithmetic mean of its Frobenius number and type. We use a procedure presented in to determine the set of all almost-symmetric numerical semigroups with given Frobenius number. In order to do this, we first calculate the set of all almost-symmetric numerical semigroups that can be constructed from an irreducible numerical semigroup.

s is an irreducible numerical semigroup. The output is the set of almost-symetric numerical semigroups that can be constructed from s by removing some of its generators as explained in ).
gap> ns := NumericalSemigroup(5,8,9,11);; gap> AlmostSymmetricNumericalSemigroupsFromIrreducible(ns); [ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup> ] gap> List(last,MinimalGeneratingSystemOfNumericalSemigroup); [ [ 5, 8, 9, 11 ], [ 5, 8, 11, 14, 17 ], [ 5, 9, 11, 13, 17 ] ]
s is a numerical semigroup. The output is true if the numerical semigroup is almost symmetric.
gap> IsAlmostSymmetricNumericalSemigroup(NumericalSemigroup(5,8,11,14,17)); true
f is an integer greater than or equal to -1. The output is the set of all almost symmetric numerical semigroups with Frobenius number f.
gap> Length(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber(12)); 15 gap> Length(IrreducibleNumericalSemigroupsWithFrobeniusNumber(12)); 2