Exploring the tree of numerical semigroups

Abstract

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g less-than-or-slanted-equals 67"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo> ⩽ </mml:mo> <mml:mn>67</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">g\leqslant 67</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and we confirm the Wilf conjecture for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g less-than-or-slanted-equals 60"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo> ⩽ </mml:mo> <mml:mn>60</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">g\leqslant 60</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .