Fromentin, Jean; Hivert, Florent Exploring the tree of numerical semigroups. (English) Zbl 1344.20075 Math. Comput. 85, No. 301, 2553-2568 (2016). 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. Cited in 25 Documents 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 Keywords:numbers of numerical semigroups; genus of numerical semigroups; trees of numerical semigroups; Frobenius numbers; algorithms Software:NumericMonoid; nsgtree; GitHub; GAP; NumericalSemigroupsWithGenus; Cilk; OEIS; numericalsgps PDFBibTeX XMLCite \textit{J. Fromentin} and \textit{F. Hivert}, Math. Comput. 85, No. 301, 2553--2568 (2016; Zbl 1344.20075) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: The number of numerical semigroups of ”genus” n; conjecturally also the number of power sum bases for symmetric functions in n variables. 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. 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