Bounds for invariants of numerical semigroups and Wilf's Conjecture

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

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