Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical semigroup tree: type-representation

Jonathan Chappelon, Jorge L. Ramírez Alfonsín, Dumitru I. Stamate

TL;DR

This work investigates the numerical semigroup tree $\mathcal{T}$ through a type-based representation, linking genus $g$ and type $t$ to the combinatorial structure of gaps and pseudo-Frobenius numbers. It introduces gap-vector encodings and the parameterization $\Lambda_{g,v}$ to study how $t$ evolves along the tree, and proves a stabilizer theorem: for large $g$ and $t$ with $t\ge (2g-1)/3$, the count $n(g,t)$ depends only on $\ell=g-t$, with two proofs and a concrete lower bound $n(g,g-\ell) \ge \ell^2-3\ell+10$ for $\ell\ge 6$. The paper also explores unimodality of the $(g,t)$-count sequences and leaves counts, provides bounds on leaf-type across genus classes, and presents conjectures about asymptotic behavior and deeper computational methods for extending the investigation. Overall, the results offer a new structural lens for numerical semigroups and suggest avenues for tackling long-standing open questions via type-driven analysis and gap-vector parametrizations.

Abstract

In this paper, we introduce a new depicting of the so-called numerical semigroup tree $\mathcal T$. By exploring computationally this improved picture, relying on the type notion of a semigroup, we found that the number of semigroups of genus $g$ and type $t$ is constante when $t$ is close to $g$ while $g$ grows. We also study the unimodality of various sequences as well as the behavior of the leaves in $\mathcal T$. We put forward several conjectures that are supported by various computational experiments.

Numerical semigroup tree: type-representation

TL;DR

This work investigates the numerical semigroup tree through a type-based representation, linking genus and type to the combinatorial structure of gaps and pseudo-Frobenius numbers. It introduces gap-vector encodings and the parameterization to study how evolves along the tree, and proves a stabilizer theorem: for large and with , the count depends only on , with two proofs and a concrete lower bound for . The paper also explores unimodality of the -count sequences and leaves counts, provides bounds on leaf-type across genus classes, and presents conjectures about asymptotic behavior and deeper computational methods for extending the investigation. Overall, the results offer a new structural lens for numerical semigroups and suggest avenues for tackling long-standing open questions via type-driven analysis and gap-vector parametrizations.

Abstract

In this paper, we introduce a new depicting of the so-called numerical semigroup tree . By exploring computationally this improved picture, relying on the type notion of a semigroup, we found that the number of semigroups of genus and type is constante when is close to while grows. We also study the unimodality of various sequences as well as the behavior of the leaves in . We put forward several conjectures that are supported by various computational experiments.

Paper Structure

This paper contains 7 sections, 11 theorems, 33 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Type-representation
  3. Some preliminaries
  4. Parent-descendants types
  5. Stabilizer sequence
  6. Unimodality and leaves
  7. Leaves

Key Result

Proposition 2.1

Let $\Lambda'$ be a numerical semigroup of genus $g>0$. Let $\Lambda=\Lambda'\cup\{F(\Lambda')\}$. Then,

Figures (2)

  • Figure 1: First levels of tree $\mathcal{T}$ where yellow, blue and red hexagons denote elements, gaps and generators greater than the Frobenius number of the corresponding semigroup respectively.
  • Figure 2: First levels of the type-representation of $\mathcal{T}$. In each level, semigroups of the same type $t$ are grouped together in a box.

Theorems & Definitions (25)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Example 3.4
  • Theorem 3.5
  • ...and 15 more