Counting Numerical Semigroups by Genus and Some Cases of a Question of Wilf

The genus of a numerical semigroup is the size of its complement. In this paper we will prove some results about counting numerical semigroups by genus. In 2008, Bras-Amorós conjectured that the ratio between the number of semigroups of genus g + 1 and the number of semigroups of genus g approaches φ, the golden ratio, as g gets large. Though several recent papers have provided bounds for counting semigroups, this conjecture is still unsolved. In this paper we will show that a certain class of semigroups, those for which twice the genus is less than three times the smallest nonzero element, grows like the Fibonacci numbers, suggesting a possible reason for this conjecture to hold. We will also verify that a 1978 question of Wilf holds for these semigroups and in certain other cases. We will also show that in several situations we can count numerical semigroups of certain genus and multiplicity by counting only semigroups of maximal embedding dimension, and that we can always interpret the number of semigroups of genus g in terms of the number of integer points in a single rational polytope. We also discuss connections with recent work of Blanco, Garćıa-Sánchez and Puerto, and mention several further open problems.