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 ≤ e n . We provide bounds for g 1 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 ≤ q e n , where q = ⌈ F + 1 g 1 ⌉ , and F + 1 ≤ e n 2 . Finally, we give an alternative, simpler proof for the Wilf conjecture if the numerical semigroup S = ⟨ g 1 , … , g e ⟩ is almost-symmetric. [ABSTRACT FROM AUTHOR]