Divsets, numerical semigroups and Wilf’s conjecture

Abstract

Let 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 divsets 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 ) .