Near-misses in Wilf’s conjecture

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.

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References

  1. 1.

    Bras-Amorós, M.: Fibonacci-like behavior of the number of numerical semigroups of a given genus. Semigroup Forum 76, 379–384 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Delgado, M.: On a question of Eliahou and a conjecture of Wilf. Math. Zeitschrift (2017). https://doi.org/10.1007/s00209-017-1902-3

  3. 3.

    Delgado, M., García-Sánchez, P.A., Morais, J.: “Numericalsgps”: a GAP package on numerical semigroups. http://www.gap-system.org/Packages/numericalsgps.html

  4. 4.

    Dobbs, D., Matthews, G.: On a question of Wilf concerning numerical semigroups. In: Focus on Commutative Rings Research. pp. 193–202. Nova Sci. Publ., New York, (2006)

  5. 5.

    Eliahou, S.: Wilf’s conjecture and Macaulay’s theorem. J. Eur. Math. Soc., (to appear). arXiv:1703.01761 [math.CO]

  6. 6.

    Fröberg, R., Gottlieb, C., Häggkvist, R.: On numerical semigroups. Semigroup Forum 35, 63–83 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Fromentin, J., Hivert, F.: Exploring the tree of numerical semigroups. Math. Comput. 85, 2553–2568 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Kaplan, N.: Counting numerical semigroups by genus and some cases of a question of Wilf. J. Pure Appl. Algebra 216, 1016–1032 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Moscariello, A., Sammartano, A.: On a conjecture by Wilf about the Frobenius number. Math. Z. 280, 47–53 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Ramírez-Alfonsín, J.L.: The diophantine Frobenius problem. In: Oxford Lecture Series in Mathematics and its Applications 30. Oxford University Press, Oxford (2005)

  11. 11.

    Rosales, J.C., García-Sánchez, P.A.: Numerical semigroups. In: Developments in Mathematics, 20. Springer, New York (2009)

  12. 12.

    Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Jiménez, J.A., Madrid, J.J.: The oversemigroups of a numerical semigroup. Semigroup Forum 67, 145–158 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Jiménez, J.A., Madrid, J.J.: Fundamental gaps in numerical semigroups. J. Pure Appl. Algebra 189, 301–313 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Sammartano, A.: Numerical semigroups with large embedding dimension satisfy Wilf’s conjecture. Semigroup Forum 85, 439–447 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Selmer, E.S.: On a linear Diophantine problem of Frobenius. J. Reine Angew. Math. 293(294), 1–17 (1977)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Sylvester, J.J.: Mathematical questions with their solutions. Educ. Times 41, 21 (1884)

    Google Scholar 

  17. 17.

    Tao, T., Vu, V.: Additive combinatorics. In: Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, (2006)

  18. 18.

    Wilf, H.: A circle-of-lights algorithm for the money-changing problem. Am. Math. Monthly 85, 562–565 (1978)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Zhai, A.: Fibonacci-like growth of numerical semigroups of a given genus. Semigroup Forum 86, 634–662 (2013)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Shalom Eliahou.

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Communicated by Fernando Torres.

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Eliahou, S., Fromentin, J. Near-misses in Wilf’s conjecture. Semigroup Forum 98, 285–298 (2019). https://doi.org/10.1007/s00233-018-9926-5

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Keywords

  • Numerical semigroup
  • Conductor
  • Apéry element
  • Sidon set
  • Additive combinatorics