On the number of fully weighted zero-sum subsequences

Abstract: Let $G$ be a finite additive abelian group with exponent $n$ and $S=g_{1}\cdots g_{t}$ be a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq{1,2,\ldots,n-1}$, let $N_{A,g}(S)$ denote the number of subsequences $T=\prod_{i\in I}g_{i}$ of $S$ such that $\sum_{i\in I}a_{i}g_{i}=g$ , where $I\subseteq\left{ 1,\ldots,t\right} $ and $a_{i}\in A$. In this paper, we prove that $N_{A,0}(S)\geq2^{|S|-D_{A}(G)+1}$, when $A=\left{ 1,\ldots,n-1\right} $, where $D_{A}(G)$ is the smallest positive integer $l$, such that every sequence $S$ over $G$ of length at least $l$ has nonempty subsequence $T=\prod_{i\in I}g_{i}$ such that $\sum_{i\in I}a_{i}g_{i}=0$, $I\subseteq\left{ 1,\ldots,t\right} $ and $a_{i}\in A$. Moreover, we classify the sequences such that $N_{A,0}(S)=2^{|S|-D_{A}(G)+1}$, where the exponent of $G$ is an odd number.