Abstract
We construct a one-to-one correspondence between a subset of numerical semigroups with genus g and γ even gaps and the integer points of a rational polytope. In particular, we give an overview to apply this correspondence to try to decide if the sequence (n g) is increasing, where n g denotes the number of numerical semigroups with genus g.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bernardini, M., Torres, F.: Counting numerical semigroups by genus and even gaps. Disc. Math. 340, 2853–2863 (2017)
Blanco, V., Puerto, J.: An application of integer programming to the decomposition of numerical semigroups. SIAM J. Discret. Math. 26(3) (2012), 1210–1237
Bras-Amorós, M.: Fibonacci-like behavior of the number of numerical semigroups of a given genus. Semigroup Forum 76, 379–384 (2008)
Fromentin, J., Hivert, F.: Exploring the tree of numerical semigroups. Math. Comp. 85(301), 2553–2568 (2016)
García-Sánchez, P.A., Rosales, J.C.: Numerical semigroups. In: Developments in Mathematics, vol. 20. Springer, New York (2009)
Kaplan, N.: Counting numerical semigroups. Am. Math. Mon. 163, 375–384 (2017)
Ramírez-Alfonsín, J.L.: The Diophantine Frobenius Problem, vol. 30. Oxford University Press, Oxford (2005)
Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Branco, M.B.: Systems of inequalities and numerical semigroups. J. Lond. Math. Soc. (2) 65, 611–623 (2002)
Rosales, J.C., García-Sánchez, P.A., García-Sánchez, J.I., Urbano-Blanco, J.M.: Proportionally modular Diophantine inequalities. J. Number Theory 103, 281–294 (2003)
Sloane, N.J.A.: The on-line encyclopedia of integer sequences, A007323 (2009). http://www.research.att.com/~njas/sequences/
Torres, F.:On γ-hyperelliptic numerical semigroups. Semigroup Forum 55, 364–379 (1997)
Zhai, A.: Fibonacci-like growth of numerical semigroups of a given genus. Semigroup Forum 86, 634–662 (2013)
Acknowledgements
The author was partially supported FAPDF-Brazil (grant 23072.91.49580.29052018). Part of this paper was presented in the “INdAM: International meeting on numerical semigroups” (2018) at Cortona, Italy. I am grateful to the referee for their comments, suggestions and corrections that allowed to improve this version of the paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Bernardini, M. (2020). Counting Numerical Semigroups by Genus and Even Gaps via Kunz-Coordinate Vectors. In: Barucci, V., Chapman, S., D'Anna, M., Fröberg, R. (eds) Numerical Semigroups . Springer INdAM Series, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-030-40822-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-40822-0_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-40821-3
Online ISBN: 978-3-030-40822-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)