A graph-theoretic approach to Wilf's conjecture

Shalom Eliahou (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

Let S ⊆ N be a numerical semigroup with multiplicity m = min(S \ {0}) and conductor c = max(N \ S) + 1. Let P be the set of primitive elements of S, and let L be the set of elements of S which are smaller than c. A longstand-ing open question by Wilf in 1978 asks whether the inequality |P||L| ≥ c always holds. Among many partial results, Wilf's conjecture has been shown to hold in case |P| ≥ m/2 by Sammartano in 2012. Using graph theory in an essential way, we extend the verification of Wilf's conjecture to the case |P| ≥ m/3. This case covers more than 99.999% of numerical semigroups of genus g ≤ 45.

Mots clés: and phrases Numerical semigroup; Apéry set; loopy graph; vertex-maximal matching; normality number; downset

Domaines: Combinatoire [math.CO]

Shalom Eliahou. A graph-theoretic approach to Wilf's conjecture. 2019. ⟨hal-02280832⟩