×

Exploring the tree of numerical semigroups. (English) Zbl 1344.20075

From the introduction: In this paper we describe an algorithm visiting all numerical semigroups up to a given genus using a well-suited representation. The interest of this algorithm is that it fits particularly well the architecture of modern computers allowing very large optimizations: we obtain the number of numerical semigroups of genus \(g\leq 67\) and we confirm the Wilf conjecture for \(g\leq 60\).
The paper is divided as follows. In Section 1 we describe the tree of numerical semigroups and give bounds for some parameters attached to a numerical semigroup. The description of our representation of numerical semigroups is done in the second section. In Section 3 we describe an algorithm based on the representation given in Section 2 and give its complexity. Section 4 is more technical and is devoted to the optimization of the algorithm introduced in Section 3. In the last section we emphasize the results obtained using our algorithm.

MSC:

20M14 Commutative semigroups
68W30 Symbolic computation and algebraic computation
20-04 Software, source code, etc. for problems pertaining to group theory
05A15 Exact enumeration problems, generating functions
05C05 Trees
11D07 The Frobenius problem
68R05 Combinatorics in computer science
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Borie, Nicolas, Generating tuples of integers modulo the action of a permutation group and applications. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), Discrete Math. Theor. Comput. Sci. Proc., AS, 767-778 (2013), Assoc. Discrete Math. Theor. Comput. Sci., Nancy · Zbl 1294.05183
[2] Bras-Amor{\'o}s, Maria, Addition behavior of a numerical semigroup. Arithmetic, geometry and coding theory (AGCT 2003), S\'emin. Congr. 11, 21-28 (2005), Soc. Math. France, Paris · Zbl 1078.20058
[3] Bras-Amor{\'o}s, Maria, Fibonacci-like behavior of the number of numerical semigroups of a given genus, Semigroup Forum, 76, 2, 379-384 (2008) · Zbl 1142.20039 · doi:10.1007/s00233-007-9014-8
[4] [Delgado] M. Delgado, Homepage, \urlhttp://cmup.fc.up.pt/cmup/mdelgado/numbers/.
[5] [NumericalSgps] M. Delgado, P. A. Garci\'a-S\'anchez, and J. Morais, NumericalSgps, A GAP package for numerical semigroups. Available via \urlhttp://www.gap-system.org. · Zbl 1365.68487
[6] [code] J. Fromentin and F. Hivert, \urlhttps://github.com/jfromentin/nsgtree.
[7] [GAP] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.7, 2015.
[8] [IntelSSE] M. Girkar, Intel instruction set architecture extensions \| Intel\textregistered developer zone, Software.intel.com, 2013.
[9] [GCCcilk] B. V. Iyer, R. Geva, and P. Halpern, Cilk\texttrademark plus in gcc, GNU Tools Cauldron, 2012.
[10] [BookDFP] J. L. Ram\'\irez Alfons\'\in, The Diophantine Frobenius problem, Oxford Lecture Series in Mathematics and its Applications, vol. 30, Oxford University Press, Oxford, 2005. · Zbl 1134.11012
[11] Rosales, J. C., Fundamental gaps of numerical semigroups generated by two elements, Linear Algebra Appl., 405, 200-208 (2005) · Zbl 1072.20070 · doi:10.1016/j.laa.2005.03.014
[12] Rosales, J. C.; Garc{\'{\i }}a-S{\'a}nchez, P. A., Numerical semigroups, Developments in Mathematics 20, x+181 pp. (2009), Springer, New York · Zbl 1220.20047 · doi:10.1007/978-1-4419-0160-6
[13] [OEIS] N. J. A. Sloane, The on-line encyclopedia of integer sequences, \urlhttp://oeis.org/. · Zbl 1274.11001
[14] [CilkIntel] Software.intel.com, Intel\textregistered Cilk\texttrademark homepage, \urlhttps://www.cilkplus.org/, 2013.
[15] [CilkRefman] Software.intel.com, Intel\textregistered Cilk\texttrademark plus reference guide, \urlhttps://software.intel.com/en-us/node/522579, 2013.
[16] [AVX2] Wikipedia, Advanced vector extension, \urlhttp://en.wikipedia.org/wiki/Advanced_Vector_Extensions, 2014.
[17] [WikipediaDuff] Wikipedia, Duff’s device, \urlhttp://en.wikipedia.org/wiki/Duff’s_device, 2014.
[18] [WikipediaHT] Wikipedia, Hyper-threading, \urlhttp://en.wikipedia.org/wiki/Hyper-Threading, 2014.
[19] [WikipediaSIMD] Wikipedia, Simd, \urlhttp://en.wikipedia.org/wiki/SIMD, 2014.
[20] [WikipediaSSE] Wikipedia, Streaming simd extensions, \urlhttp://en.wikipedia.org/wiki/Streaming_SIMD_Extensions, 2014.
[21] [WikipediaTB] Wikipedia, Turbo boost, \urlhttp://en.wikipedia.org/wiki/Intel_Turbo_Boost, 2014.
[22] Wilf, Herbert S., A circle-of-lights algorithm for the “money-changing problem”, Amer. Math. Monthly, 85, 7, 562-565 (1978) · Zbl 0387.10009
[23] Zhai, Alex, Fibonacci-like growth of numerical semigroups of a given genus, Semigroup Forum, 86, 3, 634-662 (2013) · Zbl 1276.20066 · doi:10.1007/s00233-012-9456-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.