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Near-misses in Wilf’s conjecture

  • Shalom Eliahou
  • Jean Fromentin
Research Article
  • 48 Downloads

Abstract

Let \(S \subseteq \mathbb N\) be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let \(L=S \cap [0,c-1]\) and \(W(S)=|P||L|-c\). In 1978, Herbert Wilf asked whether \(W(S) \ge 0\) always holds, a question known as Wilf’s conjecture and open since then. A related number \(W_0(S)\), satisfying \(W_0(S) \le W(S)\), has recently been introduced. We say that S is a near-miss in Wilf’s conjecture if \(W_0(S)<0\). Near-misses are very rare. Here we construct infinite families of them, with \(c=4m\) and \(W_0(S)\) arbitrarily small, and we show that the members of these families still satisfy Wilf’s conjecture.

Keywords

Numerical semigroup Conductor Apéry element Sidon set Additive combinatorics 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.EA 2597 - LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph LiouvilleUniv. Littoral Côte d’OpaleCalaisFrance
  2. 2.CNRSLilleFrance

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