Abstract
We show that the number of numerical semigroups containing two given coprime numbers \(p\) and \(q\) agrees with a quasipolynomial in \(q\) of degree exactly \(p-1\) and having constant leading coefficient lying between \(\frac{1}{(p-1)!\cdot p!}\) and \(\frac{1}{(p-1)\cdot p!}\).
Similar content being viewed by others
References
Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes–Combinatorics and Computation (Oberwolfach, 1997), DMV Sem., vol. 29, pp. 43–73, Birkhäuser, Basel (1997)
Kunz, E.: Über die Klassifikation numerischer Halbgruppen. Regensburger Mathematische Schriften, Regensburg (1987)
Kunz, E., Waldi, R.: Geometrical illustration of numerical semigroups and of some of their invariants. Semigroup Forum 89(3), 664–691 (2014)
Kunz, E., Waldi, R.: Counting numerical semigroups, arXiv:1410.7150v1 [math.CO] 27 Oct 2014
Mohanty, S.G.: Lattice Path Counting and Applications. Academic Press, New York (1979)
Moyano-Fernández, J.J., Uliczka, J.: Lattice paths with given number of turns and semimodules over numerical semigroups. Semigroup Forum 88(3), 631–646 (2014)
Rosales, J.C., Gárcia-Sánchez, P.A., Gárcia-Gárcia, J.I., Branco, M.B.: Systems of inequalities and numerical semigroups. J. Lond. Math. Soc. (2) 65(3), 611–623 (2002)
Stanley, R.P.: Enumerative Combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, vol. 49, 2nd edn. Cambridge University Press, Cambridge (2012)
Acknowledgments
We thank Helmut Knebl for leaving us his results.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Fernando Torres.
Rights and permissions
About this article
Cite this article
Hellus, M., Waldi, R. On the number of numerical semigroups containing two coprime integers \(p\) and \(q\) . Semigroup Forum 90, 833–842 (2015). https://doi.org/10.1007/s00233-015-9710-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-015-9710-8