On the type of certain numerical semigroups and a question of Wilf

Abstract

Let H be a numerical semigroup with embedding dimension $\text {edim}(H)$, type t(H), conductor c(H) and genus g(H). Wilf’s question (Am Math Mon 25:562–565, 1978) asks wether

$$\begin{aligned} \text {edim}(H)\cdot (c(H)-g(H))-c(H)\ge 0 \end{aligned}$$

for all numerical semigroups H. Positive answers in special cases have been given in Bras-Amoros (Semigroup Forum 76:379–384, 2008), Dobbs and Matthews (Focus on Commutative Rings Research, 2006), Fröberg et al. (Semigroup Forum 35:63–83, 1987), Kaplan (J. Pure Appl. Algebra 26:1016–1032 , 2012), and recently in Moscariello and Sammartano (Math Z 280:47–53, 2015). In Fröberg et al. Theorem 20 it was proved that the formula

$$\begin{aligned} (t(H)+1)\cdot (c(H)-g(H))-c(H)\ge 0 \end{aligned}$$

always holds true. Using the geometrical illustration of numerical semigroups from Kunz and Waldi (Semigroup Forum 89:664–691, 2014) it will be shown that $\text {edim}(H)\ge t(H)+1$ or even $\text {edim}(H)=t(H)+1$ for certain explicitly given collections of numerical semigroups H, which implies a positive answer to Wilf’s question for these semigroups.