## Long sequences having no two nonempty zero-sum subsequences of distinct lengths

### Volume 196 / 2020

#### Abstract

Let $G$ be an additive finite abelian group. We denote by $\mathrm {disc}(G)$ the smallest positive integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has two nonempty zero-sum subsequences of distinct lengths. In this paper, we first extend the list of the groups $G$ for which $\mathrm {disc}(G)$ is known. Then we focus on the inverse problem associated with $\mathrm {disc}(G)$. Let $\mathcal {L}_1(G)$ denote the set of all positive integers $t$ with the property that there is a sequence $S$ over $G$ with $|S|=\mathrm {disc}(G)-1$ such that all nonempty zero-sum subsequences of $S$ have the same length $t$. We determine $\mathcal {L}_1(G)$ for some special groups including the groups with large exponents compared to $|G|/\mbox {exp}(G)$, the groups of rank at most 2, the groups $C_{p^{n}}^{r}$ with $3\leq r\leq p$, and the groups $C_{mp^{n}}\oplus H$, where $H$ is a $p$-group with $\mathsf D(H)\leq p^{n}$, and $\mathsf D(H)$ denotes the Davenport constant of $H$. In particular, we find some groups $G$ with $|\mathcal {L}_1(G)|\geq 2$, which disproves a recent conjecture of W. Gao et al. [Colloq. Math. 144 (2016)]. Let $S$ be a sequence over $G$ such that all nonempty zero-sum subsequences have the same length. We determine the structure of $S$ for the cyclic group $C_n$ when $|S|\geq n+1$, and for the group $C_n\oplus C_n$ when $|S|=3n-2=\mathrm {disc}(C_n\oplus C_n)-1$.