A combinatorial problem and numerical semigroups

TL;DR

This paper presents an algorithm to compute specific sets of positive integers with particular combinatorial and numerical semigroup properties, extending understanding of their structure and relationships.

Contribution

It introduces a novel algorithmic method to identify sets satisfying complex combinatorial and numerical semigroup conditions, advancing computational approaches in this area.

Findings

Algorithm successfully computes all sets meeting the specified conditions.

Provides new insights into the structure of numerical semigroups related to the problem.

Enhances computational tools for combinatorial and algebraic investigations.

Abstract

Let $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ be two $n$-tuples of positive integers, let $X$ be a set of positive integers, and let $g$ be a positive integer. In this work we show an algorithmic process in order to compute all the sets $C$ of positive integers that fulfill the following conditions: 1) the cardinality of $C$ is equal to $g$; 2) if $x,y\in \mathbb{N} \setminus \{0\}$ and $x+y\in C$, then $C \cap \{x,y\} \neq \emptyset$; 3) if $x \in C$ and $\frac{x-b_i}{a_i} \in \mathbb{N} \setminus \{0\}$ for some $i\in \{1,\ldots,n\}$, then $\frac{x-b_i}{a_i} \in C$; 4) $X \cap C = \emptyset$.