  Let $p$ be any odd prime number. Let $k$ be any positive integer such that
$2\leq k\leq [\frac{p+1}3]+1$. Let $S = (a_1,a_2,...,a_{2p-k})$ be any sequence
in ${\Bbb Z}_p$ such that there is no subsequence of length $p$ of $S$ whose
sum is zero in $\zp$. Then we prove that we can arrange the sequence $S$ as
follows: $ S = (\underbrace{a, a, ..., a}_{u {\rm times}}, \underbrace{b, b,
>..., b}_{v {\rm times}}, a_1', a_2', >..., a_{2p-k-u-v}') $ where $u\geq v$,
$u+v\geq 2p-2k+2$ and $a-b$ generates $\zp$. This extends a result in
\cite{gao10} to all primes $p$ and $k$ satisfying $(p+1)/4+3\leq k\leq
(p+1)/3+1$. Also, we prove that if $g$ denotes the number of distinct residue
classes modulo $p$ appearing in the sequence $S$ in $\zp$ of length $2p-k$
$(2\leq k\leq [(p+1)/4]+1)$, and $g\geq 2\sqrt{2}\sqrt{k-2}$, then there exists
a subsequence of $S$ of length $p$ whose sum is zero in $\zp$.

Combinatorics;Number Theory