Divsets, numerical semigroups and Wilf’s conjecture.

AbstractLet S⊆N be a numerical semigroup with multiplicity m=min(S∖{0}) and conductor c=max(Z∖S)+1. Let P be the set of primitive elements, i.e. minimal generators, of S, and let L be the set of elements of S which are smaller than c. Wilf’s conjecture (1978) states that the inequality |P||L|≥c must hold. The conjecture has been shown to hold in case |P|≥m/2 by Sammartano in 2012, and subsequently in case |P|≥m/3 by the author in 2020. The main result in this paper is that Wilf’s conjecture holds in case |P|≥m/4 when m divides c. The proof uses <italic>divsets</italic> X, i.e. finite divisor-closed sets of monomials, as abstract models of numerical semigroups, and proceeds with estimates of the vertex-maximal matching number of the associated graph G(X). [ABSTRACT FROM AUTHOR]