An asymptotic result concerning a question of Wilf

Details An asymptotic result concerning a question of Wilf An asymptotic result concerning a question of Wilf

2011-11-11

arxiv.org/abs/1111.2779v1

Mathematics

Combinatorics

9 pages, submitted to Semigroup Forum

Scientific paper

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 $c'(\Lambda)$ to be the number of elements in $\Lambda$ less than $c(\Lambda)$. It was asked by Wilf whether $\frac{c'(\Lambda)}{c(\Lambda)} \ge \frac{1}{e(\Lambda)}$ 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 $\frac{c'(\Lambda)}{c(\Lambda)} \ge \frac{1}{k} - \epsilon$ holds for all but finitely many numerical semigroups $\Lambda$ satisfying $e(\Lambda) = k$.

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