# Classifications of $\ell$-Zero-Sumfree Sets

## TL;DR

This paper introduces an algorithm to compute the simplicial complex of $ ext{l}$-zero-sumfree subsets in cyclic groups, and provides theoretical results for specific parameter families, including lattice and polynomial invariants.

## Contribution

It develops a novel algorithm based on $(n, ext{l})$-congruent partitions and determines the structure of $ ext{l}$-zero-sumfree complexes for certain infinite families.

## Key findings

- Algorithm for computing $ ext{l}$-zero-sumfree complexes
- Explicit descriptions for infinite parameter families
- Computed intersection lattices and characteristic polynomials

## Abstract

The set of all $\ell$-zero-sumfree subsets of $\mathbb{Z}/n\mathbb{Z}$ is a simplicial complex denoted by $\Delta_{n,\ell}$ We create an algorithm via defining a set of integer partitions we call $(n,\ell)$-congruent partitions in order to compute this complex for moderately-sized parameters $n$ and $\ell$. We also theoretically determine $\Delta_{n,\ell}$ for several infinite families of parameters, and compute the intersection lattices and the characteristic polynomials of the corresponding coordinate subspace arrangements.