On the seeds and the great-grandchildren of a numerical semigroup

TL;DR

This paper revisits and improves the seeds algorithm for exploring the semigroup tree, introducing a new definition, analyzing seeds of small semigroups, and efficiently finding great-grandchildren, enabling exploration of larger genera and proving cases of the Wilf conjecture.

Contribution

The paper introduces a new, more manageable definition of seeds, optimizes the seeds algorithm with bitwise operations, and applies it to explore the semigroup tree up to genus 66, advancing the understanding of numerical semigroups.

Findings

The seeds algorithm outperforms RGD for genera up to half the maximum integer size.

Proved no Eliahou semigroups of genus 66, confirming the Wilf conjecture up to that genus.

Discovered three Eliahou semigroups of genus 67, including a new family.

Abstract

We present a revisit of the seeds algorithm to explore the semigroup tree. First, an equivalent definition of seed is presented, which seems easier to manage. Second, we determine the seeds of semigroups with at most three left elements. And third, we find the great-grandchildren of any numerical semigroup in terms of its seeds. The RGD algorithm is the fastest known algorithm at the moment. But if one compares the originary seeds algorithm with the RGD algorithm, one observes that the seeds algorithm uses more elaborated mathematical tools while the RGD algorithm uses data structures that are better adapted to the final C implementations. For genera up to around one half of the maximum size of native integers, the newly defined seeds algorithm performs significantly better than the RGD algorithm. For future compilators allowing larger native sized integers this may constitute a powerful tool to explore the semigroup tree up to genera never explored before. The new seeds algorithm uses bitwise integer operations, the knowledge of the seeds of semigroups with at most three left elements and of the great-grandchildren of any numerical semigroup, apart from techniques such as parallelization and depth first search as wisely introduced in this context by Fromentin and Hivert. The algorithm has been used to prove that there are no Eliahou semigroups of genus $66$, hence proving the Wilf conjecture for genus up to $66$. We also found three Eliahou semigroups of genus $67$. One of these semigroups is neither of Eliahou-Fromentin type, nor of Delgado's type. However, it is a member of a new family suggested by Shalom Eliahou.