Bounds for invariants of numerical semigroups and Wilf's Conjecture

Given coprime positive integers $g_1<\dots <g_e$, the Frobenius number $F=F(g_1,\dots ,g_e)$ is the largest integer not representable as a linear combination of $g_1,\dots ,g_e$ with non-negative integer coefficients. Let $n$ denote the number of all representable non-negative integers less than $F$; Wilf conjectured that $F+1\le en$. We provide bounds for $g_1$ and for the type of the numerical semigroup $S=⟨g_1,\dots ,g_e⟩$ in function of $e$ and $n$, and use these bounds to prove that $F+1\le qen$, where $q=\lceil \frac{F+1}{g_1}\rceil$, and $F+1\le e{n}^2$. Finally, we give an alternative, simpler proof for the Wilf conjecture if the numerical semigroup $S=⟨g_1,\dots ,g_e⟩$ is almost-symmetric.