Semigroups with fixed multiplicity and embedding dimension

TL;DR

This paper explores packed numerical semigroups with fixed multiplicity and embedding dimension, using them to partition the set of such semigroups and to investigate Wilf's conjecture within this framework.

Contribution

It introduces the concept of packed numerical semigroups to analyze and partition semigroups with fixed parameters, and examines Wilf's conjecture in this context.

Findings

Partition of semigroups with fixed multiplicity and embedding dimension.

Verification of Wilf's conjecture in specific cases.

Structural insights into packed numerical semigroups.

Abstract

Given $m\in \mathbb{N},$ a numerical semigroup with multiplicity $m$ is called packed numerical semigroup if its minimal generating set is included in $\{m,m+1,\ldots, 2m-1\}.$ In this work, packed numerical semigroups are used to built the set of numerical semigroups with fixed multiplicity and embedding dimension, and to create a partition in this set. Moreover, Wilf's conjecture is checked in the tree associated to some packed numerical semigroups.