论文标题
固定frobenius数的数值半群中属的分布
Distribution of genus among numerical semigroups with fixed Frobenius number
论文作者
论文摘要
数值半群是具有有限补体的自然数的子序列。其补体的大小称为该属，补体中最大的数字称为其Frobenius数字。我们考虑具有固定的Frobenius数字$ f $并分析其属的数值半群。我们在这组数字半群中发现了属的渐近分布，并表明它是高斯和功率序列的产物。我们表明，几乎所有具有Frobenius数字$ f $的数值半群都将属属接近$ \ frac {3f} {4} $。我们表示Frobenius数字$ f $ by $ n（f）$的数值半群。虽然$ n（f）$不是单调的，但我们证明每$ f $ $ n（f）<n（f+2）$。
A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number $f$ and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number $f$ have genus close to $\frac{3f}{4}$. We denote the number of numerical semigroups of Frobenius number $f$ by $N(f)$. While $N(f)$ is not monotonic we prove that $N(f)<N(f+2)$ for every $f$.
