A Partition of the Set of Numerical Semigroups Associated to Wilf's Conjecture

Abstract

If S is a numerical semigroup, we denote by n(S) the cardinality of N(S) = {s ∈ S | s < F(S)}, F(S) = max(Z\S) and by g(S) the cardinality of N\S. Let q ∈ Q, q ≥ 1 and {k, F} ⊆ N\{0}. In this paper we introduce the sets B(q) = {S | S is a numerical semigroupand g(S) n(S) = q} and A (k, F) = {S ∈ A (k) | F(S) = F}. The Wilf’s conjecture will be reformulated by these sets. Also we show two algorithms which compute the elements of the sets A (k, F) = {S ∈ A (k) | F(S) = F} and B(q, k) = {S | S is a numerical semigroup, g(S) = ak and n(S) = bk}.