Near-misses in Wilf's conjecture

Shalom Eliahou (1) , Jean Fromentin (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

Let S ⊆ N be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let L = S ∩ [0, c − 1] and W(S) = |P||L| − c. In 1978, Herbert Wilf asked whether W(S) ≥ 0 always holds, a question known as Wilf's conjecture and open since then. A related number W0(S), satisfying W0(S) ≤ W(S), has recently been introduced. We say that S is a near-miss in Wilf's conjecture if W0(S) < 0. Near-misses are very rare. Here we construct infinite families of them, with c = 4m and W0(S) arbitrarily small, and we show that the members of these families still satisfy Wilf's conjecture.

Mots clés: Numerical semigroup; conductor; Apéry element; Sidon set; additive combinatorics

Domaines: Combinatoire [math.CO]

Shalom Eliahou, Jean Fromentin. Near-misses in Wilf's conjecture. 2017. ⟨hal-01613610⟩