Counting the Ideals with a Given Genus of a Numerical Semigroup with Multiplicity Two

Abstract

Let S and T be two numerical semigroups. We say that T is an I(S)-semigroup if T\{0} is an ideal of S. Given k a positive integer, we denote by ∆(k) the symmetric numerical semigroup generated by {2, 2k + 1}. In this paper we present a formula which calculates the number of I(S)- semigroups with genus g(∆(k)) + h for some nonnegative integer h and which we will denote by i(∆(k), h). As a consequence, we obtain that the sequence {i(∆(k), h)}h∈N is never decreasing. Besides, it becomes stationary from a certain term.