On sets related to integer partitions with quasi-required elements and disallowed elements

TL;DR

This paper investigates the structure of certain sets related to integer partitions, focusing on minimal sets avoiding specific linear combinations and ensuring partition conditions, by translating the problem into the framework of numerical semigroups.

Contribution

It introduces a novel approach to characterize minimal sets linked to integer partitions using numerical semigroups theory.

Findings

Characterization of minimal sets avoiding generated elements

Connection established between partition conditions and numerical semigroups

Method for computing such sets based on semigroup properties

Abstract

Given a set A of non-negative integers and a set B of positive integers,we are interested in computing all sets C (of positive integers) that are minimal in the family of sets K (of positive integers) such that (i) K contains no elements generated by non-negative integer linear combinations of elements in A and (ii) for any partition of an element in B there is at least one summand that belongs to K. To solve this question, we translate it into a numerical semigroups problem.