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 g F+1−g ≤ t ≤ 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 = 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. The number of numerical semigroups with genus g and type g − α is also eventually constant for large g.

中文翻译：

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小型和大型数值半群

数值半群是具有有限补码的自然数的子半群。数值半群的一些关键性质是它的 Frobenius 数F, 属G并输入吨. 已知对于任何数值半群 G F+1-G ≤ 吨 ≤ 2G - F. 数值半群吨 = 2G - F被称为几乎对称，我们引入了一个表征它们的新属性。我们给出了数值半群的明确表征吨 = G F+1-G. 我们证明对于一个固定的α具有 Frobenius 数的数值半群的数量F并输入F - α最终是恒定的大F. 有属的数值半群的数目G并输入G - α对于大的也是最终恒定的G.