On Sequences Without Short Zero-Sum Subsequences

Let $G$ be a finite abelian group. It is well known that every sequence $S$ over $G$ of length at least $|G|$ contains a zero-sum subsequence of length at most $\mathsf{h}(S)$, where $\mathsf{h}(S)$ is the maximal multiplicity of elements occurring in $S$. It is interesting to study the corresponding inverse problem, that is to find information on the structure of the sequence $S$ which does not contain zero-sum subsequences of length at most $\mathsf{h}(S)$. Under the assumption that $|\sum(S)|< \min\{|G|,2|S|-1\}$, Gao, Peng and Wang showed that such a sequence $S$ must be strictly behaving. In the present paper, we explicitly give the structure of such a sequence $S$ under the assumption that $|\sum(S)|=2|S|-1<|G|$.