Wilf's conjecture for numerical semigroups

Let $S\subseteq \mathbb{N}$ be a numerical semigroup with multiplicity $m$, embedding dimension $\nu$ and conductor $c=f+1=qm-\rho$ for some $q,\rho \in \mathbb{N}$ with $\rho <m$. Let Ap$(S,m) = {w_0<w_1 < \ldots < w_{m-1}}$betheAp\acute{e}rysetof$S$. The aim of this paper is to prove Wilf’s Conjecture in some special cases. First, we prove that if $w_{m-1}\geq w_1+w_\alpha$and$(2+\frac{\alpha-3}{q})\nu\geq m$forsome$1<\alpha<m-1$, then $S$ satisfies Wilf’s Conjecture. Then, we prove the conjecture in the following cases: $(2+\frac{1}{q})\nu \ge m$, $m-\nu \le 5$ and $m=9$. Finally, the conjecture is proved if $w_{m-1}\ge w_{\alpha -1}+w_{\alpha }$ and $(\frac{\alpha +3}{3})\nu \ge m$ for some $1<\alpha <m-1$.