Exploring the tree of numerical semigroups

Jean Fromentin (1) , Florent Hivert (2, 3)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
2 LITIS - Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes
3 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)

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 67 and we confirm the Wilf conjecture for g 60.

Keywords: semigroups; tree; algorithm; SSE; Cilk; optimization

Domains: Mathematics [math]; Combinatorics [math.CO]; Mathematics [math]; Group Theory [math.GR]

Jean Fromentin, Florent Hivert. Exploring the tree of numerical semigroups. Mathematics of Computation, 2016, 85, pp.2553-2568 ⟨10.1090/mcom/3075⟩. ⟨hal-00823339v3⟩