Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T11:48:55.566Z Has data issue: false hasContentIssue false
  • English
  • Français

A problem of integer partitions and numerical semigroups

Part of: Additive number theory; partitions Combinatorics Semigroups

Published online by Cambridge University Press:  27 December 2018

J. C. Rosales  and
M. B. Branco
J. C. Rosales
Affiliation:
Departamento de Álgebra, Universidad de Granada, E-18071 Granada, Spain ([email protected])
M. B. Branco
Affiliation:
Departamento de Matem1ática, Universidade de Évora, 7000 Évora, Portugal ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let C be a set of positive integers. In this paper, we obtain an algorithm for computing all subsets A of positive integers which are minimals with the condition that if x1 + … + xn is a partition of an element in C, then at least a summand of this partition belongs to A. We use techniques of numerical semigroups to solve this problem because it is equivalent to give an algorithm that allows us to compute all the numerical semigroups which are maximals with the condition that has an empty intersection with the set C.

Keywords

integer partitionsnumerical semigroupsirreducibles numerical semigroupsApéry set

MSC classification

Primary: 05A17: Partitions of integers
Secondary: 11P81: Elementary theory of partitions 20M14: Commutative semigroups
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 969 - 978
Copyright
Copyright © Royal Society of Edinburgh 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Andrews, G. E.. The theory of partitions, encyclopedia of mathematics and its applications, vol. 2, first published 1976 by Addison-Wesley Publishing Company. (Reissued: Cambridge University Press, Cambridge, 1985 and 1998).Google Scholar
2Apéry, R.. Sur les branches superlinéaires des courbes algébriques. C. R. Acad. Sci. Paris 222 (1946), 1198–2000.Google Scholar
3Barucci, V., Dobbs, D. E. and Fontana, M.. Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains. Memoirs of the Amer. Math. Soc. 598 (1997), 78pp.Google Scholar
4Blanco, V. and Rosales, J. C.. The tree of irreducible numerical semigroups with fixed Frobenius number. Forum Math. 25 (2013), 1249–1261.Google Scholar
5Delgado, M., García-Sánchez, P. A. and Morais, J.. Numericalsgps: a gap package on numerical semigroups, (http://www.gap-system.org/Packages/numericalsgps.html).Google Scholar
6Hardy, G. H. and Ramanujan, S.. Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (3e éd.) 17 (1918), 75–115.Google Scholar
7Jiménez Urroz, J.. De partitione numerorum. LA Gaceta RSME 5 (2002), 443–454.Google Scholar
8Ono, K.. The distribuition of the partition function modulo m. Ann. Math. 151 (2000), 293–307.Google Scholar
9Ramírez Alfonsín, J. L.. The diophantine Frobenius problem (New York: Oxford Univ. Press, 2005).Google Scholar
10Rosales, J. C. and Branco, M. B.. Irreducible numerical semigroups. Pacific J. Math. 209 (2003), 131–143.Google Scholar
11Rosales, J. C. and García-Sánchez, P. A.. Numerical semigroups. Developments in Mathematics, vol. 20 (New York: Springer, 2009).Google Scholar
12Rosales, J. C., García-Sánchez, P. A., García-García, J. I. and Jimenez-Madrid, J. A.. The oversemigroups of a numerical semigroup. Semigroup Forum 67 (2003), 145–158.Google Scholar