Generating functions for the quotients of numerical semigroups

Abstract

We propose a class of generating functions denoted by ${\text{RGF}}_p(x)$, which is related to the Sylvester denumerant for the quotients of numerical semigroups. Using MacMahon's partition analysis, we can obtain ${\text{RGF}}_p(x)$ by extracting the constant term of a rational function. We use ${\text{RGF}}_p(x)$ to give a system of generators of the quotient of the numerical semigroup $⟨a_1,a_2,a_3⟩$ by $p$ for a small positive integer $p$ and we characterise the generators for $\frac{⟨A⟩}{p}$ for a general numerical semigroup $A$ and any positive integer $p$.

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• A Euclid style algorithm for MacMahon's partition analysis
• The Frobenius number for sequences of triangular numbers associated with number of solutions
• Numerical semigroups problem list
• Quotients of numerical semigroups generated by two numbers