Numerical Semigroups of small and large type

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 $\frac{g}{F+1-g}\leq t\leq 2g-F$. Numerical semigroups with $t=2g-F$ are called almost symmetric, we introduce a new property that characterises them. We give an explicit characterisation of numerical semigroups with $t=\frac{g}{F+1-g}$. We show that for a fixed $α$ the number of numerical semigroups with Frobenius number $F$ and type $F-α$ is eventually constant for large $F$. Also the number of numerical semigroups with genus $g$ and type $g-α$ is also eventually constant for large $g$.

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