Exploring the tree of numerical semigroups
Jean Fromentin (LMPA), Florent Hivert (LITIS)
TL;DR
This paper introduces an efficient algorithm for enumerating all numerical semigroups up to a certain genus, enabling large-scale computations and verification of conjectures in the field.
Contribution
The paper presents a novel algorithm optimized for modern computers to systematically explore numerical semigroups up to high genus values.
Findings
Counted numerical semigroups up to genus 67
Confirmed Wilf conjecture for genus up to 60
Demonstrated algorithm's efficiency and scalability
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 g 67 and we confirm the Wilf conjecture for g 60.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Polynomial and algebraic computation · Advanced Combinatorial Mathematics