On the enumeration of the set of numerical semigroups with fixed Frobenius number

In this paper, we present an efficient algorithm to compute the whole set of numerical semigroups with a given Frobenius number FF. The methodology is based on the construction of a partition of that set by a congruence relation. It is proven that each class in the partition contains exactly one irreducible and one homogeneous numerical semigroup, and from those two elements the whole class can be reconstructed. An alternative encoding of a numerical semigroup, its Kunz-coordinates vector, is used to propose a simple methodology to enumerate the desired set by manipulating a lattice polytope of 0–1 vectors and solving certain integer programming problems over it.