Abstract: A numerical semigroup S is an additive submonoid of N whose complement is finite. The cardinality of N \ S is called the genus of S and is denoted by g(S). The first nonzero element of S is called the multiplicity of S and is denoted by m(S). In this talk, we will introduce these ideas, explain their connection to the Frobenius problem, and mention some interesting results. We will then discuss our contribution to the field.

In our work, we focus on the number N (m, g) of numerical semigroups with parameters m and g. It is known that N (m, g) can be formulated as the number of integer points in a certain family of rational polytopes and hence coincides with a quasi-polynomial in g of degree m − 2. We show that the leading coefficient is constant and provide an interpretation for it. Moreover, we relate N (m, g) to the number M E D (m,g) of maximally embedded numerical semigroups with the same parameters, hence proving a conjecture of Kaplan. We conclude by discussing the special case m = 4.