Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 595–627 | Cite as

On a question of Eliahou and a conjecture of Wilf

  • Manuel Delgado


To a numerical semigroup S, Eliahou associated a number \({{\mathrm{E}}}(S)\) and proved that numerical semigroups for which the associated number is non negative satisfy Wilf’s conjecture. The search for counterexamples for the conjecture of Wilf is therefore reduced to semigroups which have an associated negative Eliahou number. Eliahou mentioned 5 numerical semigroups whose Eliahou number is \(-1\). The examples were discovered by Fromentin who observed that these are the only ones with negative Eliahou number among the over \(10^{13}\) numerical semigroups of genus up to 60. We prove here that for any integer n there are infinitely many numerical semigroups S such that \({{\mathrm{E}}}(S)=n\), by explicitly giving families of such semigroups. We prove that all the semigroups in these families satisfy Wilf’s conjecture, thus providing not previously known examples of semigroups for which the conjecture holds.


Numerical semigroup Wilf conjecture Computational mathematics 

Mathematics Subject Classification

20M14 05A20 11B75 20-04 



Besides the formal acknowledgement to the Centre for Mathematics of the University of Porto (CMUP) made somewhere in this paper, I would like to thank CMUP for providing me the computational tools necessary to this kind of works. I would like also to thank Shalom Eliahou for his suggestions and comments on preliminary versions. Special thanks to Shalom for encouraging me to publish this work. The anonimous referee, who carefully read the manuscript and gave me the appropriate remarks, deserves also my special ackowledgement.


  1. 1.
    Bras-Amorós, M.: Fibonacci-like behaviour of the number of numerical semigroups of a given genus. Semigroup Forum 76, 379–384 (2008)Google Scholar
  2. 2.
    Delgado, M.: “ intpic”, A GAP Package for Drawing Integers, Version 0.2.1 (2015).
  3. 3.
    Delgado, M., García-Sánchez, P.A., Morais, J.: “ NumericalSgps”, A GAP Package for Numerical Semigroups, Version 1.0.1 (2015).
  4. 4.
    Eliahou, S.: Wilf’s conjecture and Macaulay’s theorem. arXiv:1703.01761 [math.CO]
  5. 5.
    Fromentin, J., Hivert, F.: Exploring the tree of numerical semigroups. Math. Comput. 85, 2553–2568 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups. In: Developments in Mathematics, vol. 20. Springer (2010)Google Scholar
  7. 7.
    The GAP Group: GAP—groups, algorithms, and programming, Version 4.8.7 (2017).
  8. 8.
    Wilf, H.: A circle-of-lights algorithm for the money-changing problem. Am. Math. Monthly 85, 562–565 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhai, A.: Fibonacci-like growth of numerical semigroups of a given genus. Semigroup Forum 86, 634–662 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.CMUP, Departamento de Matemática, Faculdade de CiênciasUniversidade do PortoPortoPortugal

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