Geometrical illustration of numerical semigroups and of some of their invariants

Let \(p,q\in \mathbb {N}\) be relatively prime integers with \(1 . In Knebl et al. (J Algebra 348:315–335, 2011) the numerical semigroups containing \(p\) and \(q\) have been illustrated by certain lattice paths in the plane. In this paper we derive some further information from this visualization, for example formulas to compute or estimate the number of certain semigroups by counting lattice paths, and formulas for their genus and Frobenius number.