Classifications of $\ell$-Zero-Sumfree Sets

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.