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 [Formula: see text], then we denote by i(S,k) the number of I(S)-semigroups with genus g(S) + k. In this work, we conjecture that [Formula: see text] if [Formula: see text], and we show that there is a term from which this sequence becomes stationary. That is, there exists [Formula: see text] such that i(S, k˙S)= i(S, k˙S + h) for all [Formula: see text] Moreover, we prove that the conjecture is true for ordinary numerical semigroups, that is, numerical semigroups which the form [Formula: see text] for some positive integer. Additionally, we calculate the term from which the sequence becomes stationary.