Bounds for invariants of numerical semigroups and Wilf's conjecture

Given coprime positive integers g(1) < ... < g(e), the Frobenius number F = F(g(1),..., g(e)) is the largest integer not representable as a linear combination of g(1),..., 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 <= en. We provide bounds for g1 and for the type of the numerical semigroup S = < g(1),..., g(e)> in function of e and n, and use these bounds to prove that F + 1 <= qen, where q = [F+1/g1], and F + 1 <= en(2). Finally, we give an alternative, simpler proof for theWilf conjecture if the numerical semigroup S = < g(1),..., g(e)> is almost-symmetric.