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{\bf Gerard Jennhwa Chang, Sheng-Hua Chen, Yongke Qu, Guoqing Wang and Haiyan Zhang}
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{\bf On the Number of Subsequences with a Given Sum in a Finite Abelian Group}
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Suppose $G$ is a finite abelian group and $S$ is a sequence of
elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the
number of subsequences of $S$ with sum $g$. The purpose of this paper
is to investigate the lower bound for $N_g(S)$.  In particular, we
prove that either $N_g(S)=0$ or $N_g(S)\ge2^{|S|-D(G)+1}$, where
$D(G)$ is the smallest positive integer $\ell$ such that every
sequence over $G$ of length at least $\ell$ has a nonempty zero-sum
subsequence. We also characterize the structures of the extremal
sequences for which the equality holds for some groups.



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