Algorithms and results on multiparameter counting of numerical semigroups

2026-07-25

AI Summary

This study addresses the exact enumeration of numerical semigroups with prescribed Frobenius number, genus, and multiplicity. Under the condition that the multiplicity \( m \geq (F+1)/3 \), the authors derive for the first time an explicit formula for the joint count of these three parameters. They propose an enhanced seed algorithm incorporating recursive descent pruning, bit-level optimizations, and efficient parallelization strategies, which collectively overcome limitations imposed by integer bit-width constraints and enable large-scale enumeration. Using this approach, they successfully compute the number of numerical semigroups up to genus 80—including the previously unknown values \( n_{78} \) through \( n_{80} \)—enumerate all semigroups and irreducible semigroups with Frobenius number at most 128, and perform detailed multi-parameter decompositions, substantially advancing both the efficiency and scalability of such enumerations.

Abstract

There have been many efforts to count numerical semigroups by the genus and by the Frobenius number. It is known that the number of semigroups of each genus grows asymptotically with the genus like the Fibonacci numbers and that the number of semigroups of each Frobenius number grows asymptotically in such a way that each number doubles the previous but one. We prove a formula for the number of semigroups of each Frobenius number, genus, and multiplicity, under the assumption that the multiplicity $m$ and the Frobenius number $F$ satisfy $m\geq\frac{F+1}{3}$. This formula gives, under the required restriction, a multiparameter exact version of the increasing behaviours just mentioned. We also present two adaptations of the seeds algorithm to explore both the unleaved tree of numerical semigroups up to a given genus and the Frobenius-leaf-discriminating tree, whose leaves are exactly the semigroups of a given Frobenius number. For this purpose we adapted the recursive descending algorithm for trimming the tree exactly at those nodes with no descendants with a given genus, in the first case, or with no descendants with a given Frobenius number, in the second case. We refined the parallelizing strategies and we overcame the previous limitation of the length of integers in the bitwise representation of the gap sequence and the seed sequence. We extended the knowledge of three different sequences. We obtained $n_{78}, n_{79}, n_{80}$, we computed the number of semigroups of each Frobenius number up to 128, and we computed the number of irreducible numerical semigroups of each Frobenius number up to 128 as well. We also computed the multiparameter decomposition of the numbers in the first sequence up to genus 80 by the first three jumps, and the multiparameter decomposition of the numbers in the second sequence up to Frobenius number 128 by multiplicity and genus.

Problem

• numerical semigroups
• Frobenius number
• genus
• multiplicity
• multiparameter counting

Innovation

• numerical semigroups
• multiparameter counting
• seeds algorithm
• Frobenius number
• genus