Narrow Results

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number $F$, genus $g$ and type $t$. It is known that for any numerical semigroup $\frac{g}{F+1-g}\le t\le 2g-F$. Numerical semigroups with $t=2g-F$ are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with $t=\frac{g}{F+1-g}$. We show that for a fixed $\alpha$ the number of numerical semigroups with Frobenius number $F$ and type $F-\alpha$ is eventually constant for large $F$. The number of numerical semigroups with genus $g$ and type $g-\alpha$ is also eventually constant for large $g$.