An asymptotic result concerning a question of Wilf

Abstract

Let $\Lambda$ be a numerical semigroup with embedding dimension $e(\Lambda )$. Define $c(\Lambda )$ to be one plus the largest integer not in $\Lambda$, and define $c^{\prime }(\Lambda )$ to be the number of elements in $\Lambda$ less than $c(\Lambda )$. It was asked by Wilf whether $\frac{c^{\prime }(\Lambda )}{c(\Lambda )}\ge \frac{1}{e(\Lambda )}$ always holds. We prove an asymptotic version of this conjecture: we show that for a fixed positive integer $k$ and any $\epsilon >0$, the inequality $\frac{c^{\prime }(\Lambda )}{c(\Lambda )}\ge \frac{1}{k}-\epsilon$ holds for all but finitely many numerical semigroups $\Lambda$ satisfying $e(\Lambda )=k$.

9 pages, submitted to Semigroup Forum

References in corpus (1)

• Computing the number of numerical semigroups using generating functions

Cited by in corpus (4)

• Counting Numerical Semigroups
• Variants on a question of Wilf
• Note on a question of Wilf
• Distributions of weights and a question of Wilf