On the number of fully weighted zero-sum subsequences

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 $A\subseteq \{1,2,\dots ,n-1\}$, 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$\mathrm{.}Inthispaper,weprovethat$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 $N_{A,0}(S)=2^{\mathrm{\mid }S\mathrm{\mid }-D_A(G)+1}$, where the exponent of $G$ is an odd number.