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 $A\subseteq \{1,2,\dots ,n-1\}$, 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 $I\subseteq \left\{1,\dots ,t\right\}$ and $a_i\in A$. In this paper, we prove that $N_{A,0}(S)\ge 2^{|S|-D_A(G)+1}$, when $A=\left\{1,\dots ,n-1\right\}$, 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 ${\sum }_{i\in I}{a}_i{g}_i=0$, $I\subseteq \left\{1,\dots ,t\right\}$ 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.