Counting the ideals with given genus of a numerical semigroup

Abstract

If S is a numerical semigroup, denote by g(S) the genus of S. A numerical semigroup T is an I(S)-semigroup if T\{0} is an ideal of S. If k ∈ N, then we denote by i(S, k) the number of I(S)-semigroups with genus g(S) + k. In this work, we conjecture that i(S, a) ≤ i(S, b) if a ≤ b, and we show that there is a term from which this sequence becomes stationary. That is, there exists kS ∈ N such that i(S, kS) = i(S, kS + h) for all h ∈ N. Moreover, we prove that the conjecture is true for ordinary numerical semigroups, that is, numerical semigroups which the form {0,m,→} for some positive integer. Additionally, we calculate the term from which the sequence becomes stationary.