On the Enumeration of the Set of Elementary Numerical Semigroups with Fixed Multiplicity, Frobenius Number or Genus


Download PDF

Authors: J. C. ROSALES AND M.. BRANCO

DOI: 10.46793/KgJMat2203.433R

Abstract:

In this paper we give algorithms that allow to compute the set of every elementary numerical semigroups with given genus, Frobenius number and multiplicity. As a consequence we obtain formulas for the cardinality of these sets.



Keywords:

Elementary numerical semigroups, Fibonacci sequence, genus, Frobenius number and multiplicity.



References:

[1]   J. Backelin, On the number of semigroups of natural numbers, Math. Scand. 66(2) (1990), 197–215.

[2]   V. Blanco, P. A. García-Sánchez and J. Puerto, Counting numerical semigroups with short generating functions, Internat. J. Algebra Comput. 21(7) (2011), 1217–1235.

[3]   V. Blanco and J. C. Rosales, The set of numerical semigroups of a given genus, Semigroup Forum 85 (2012), 255–267.

[4]   M. Bras-Amorós, Bounds on the number of numerical semigroups of a given genus, J. Pure Appl. Algebra 213(6) (2008), 997–1001.

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

[6]   S. Elizalde, Improved bounds on the number of numerical semigroups of a given genus, J. Pure Appl. Algebra 214(10) (2010), 1862–1873.

[7]   N. Kaplan, Couting numerical semigroups by genus and some cases a question of Wilf, J. Pure Appl. Algebra 216(5) (2012), 1016–1032.

[8]   N. Kaplan and L. Ye, The proportion of Weierstrass semigroups, J. Algebra 373 (2013), 377–391.

[9]   J. L. Ramirez Alfonsín, The Diophantine Forbenius Problem, Oxford University Press, London, 2005.

[10]   A. M. Robles-Pérez and J. C. Rosales, The numerical semigroups of phases’lengths in a simple alphabet, The Scientific World Journal 2013 (2013), paper ID 459024, 9 pages.

[11]   J. C. Rosales, Families of numerical semigroups closed under finite intersections and for the Frobenius number, Houston J. Math. 34 (2008), 339–348.

[12]   J. C. Rosales and P. A. García-Sánchez, Numerical Semigroups, Developments in Mathematics 20, Springer, New York, 2009.

[13]   Y. Zhao, Constructing numerical semigroups of a given genus, Semigroup Forum 80(2) (2009), 242–254.