On the number of numerical semigroups containing two coprime integers \(p\) and \(q\).
The manuscript under review relates numerical semigroups containing two coprime integers with rational convex polytopes and with lattice paths in the positive orthant of the plane. This enables the authors not only to determine the number of these objects, but also describe the shape of the function that counts them. Let \(p\) and \(q\) be coprime integers and denote by \(n(p,q)\) the number of numerical semigroups containing \(\{p,q\}\). Then fixing \(p\), there is a quasipolynomial of degree \(p-1\) taking the value \(n(p,q)\) for each \(q\in\mathbb N\) (the set of nonnegative integers), and the leading coefficient is constant. Moreover, bounds for the leading coefficient are given, and the value of \(n(p,q)\) is increasing in both variables. A numerical semigroup is a submonoid of \((\mathbb N,+)\) with finite complement in \(\mathbb N\). Having finite complement in \(\mathbb N\) is equivalent to having greatest common divisor of its elements equal to one. If we are given \(p\) and \(q\) coprime, then the submonoid they span is a numerical semigroup. Since the complement of this numerical semigroup is finite, there are only finitely many numerical semigroups containing \(p\) and \(q\); this is why \(n(p,q)\) is finite. The authors associate to each numerical semigroup containing \(p\) and \(q\) a point with nonnegative integer coordinates in a cone in \(\mathbb R^p\) intersected by the hyperplane \(x_p=q\). This is done via Apéry sets with respect to \(p\). Then they use Erhart's results to show that the number of integer points in that polytope is a quasipolynomial. In order to prove that the function is increasing in both variables and the properties of the leading coefficient, the authors relate to each point in the above polytope a lattice path in the plane from \((0,p)\) to \((q,0)\) with unit steps to the right and down. Examples, figures and explicit computations are given to illustrate the results.
scientific article; zbMATH DE number 6493128
scientific article