On the Number of Weighted Zero-sum Subsequences

Let $G$ be a finite additive abelian group with exponent $d^{k}n,d,n>1,$ and $k$ a positive integer. For $S$ a sequence over $G$ and $A=\{1,2,\dots ,d^{k}n-1\}\setminus \{d^{k}n\mathrm{/}{d}^i:i\in [1,k]\},$ we investigate the lower bound of the number $N_{A,0}(S)$, which denotes the number of $A$-weighted zero-sum subsequences of $S\mathrm{.}$ In particular, we prove that $N_{A,0}(S)\ge 2^{\mathrm{\mid }S\mathrm{\mid }-D_A(G)+1},$ where $D_A(G)$ is the $A$-weighted Davenport Constant. We also characterize the structures of the extremal sequences for which equality holds for some groups.