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{\bf Artur Szyma\'nski and A. Pawe{\l} Wojda}
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{\bf Cyclic partitions of complete uniform hypergraphs}
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By $K^{(k)}_n$ we denote the complete $k$-uniform hypergraph of order $n$,
$1\le k \le n-1$, i.e. the hypergraph with the set $V_n=\{
1,2,...,n\}$ of vertices and the set $V_n \choose k$ of edges. If
there exists a permutation $\sigma$ of the set $V_n$ such that $\{
E,\sigma (E),..., \sigma^{q-1}(E) \}$ is a partition of the set $V_n
\choose k$ then we call it cyclic $q$-partition of $K^{(k)}_n$ and $\sigma$
is said to be a $(q,k)$-complementing. \\
In the paper, for arbitrary integers $k,q$ and $n$, we give a
necessary and sufficient condition for
a permutation to be $(q,k)$-complementing permutation of $K^{(k)}_n$.\\
By $\tilde{K}_n$ we denote the hypergraph with the set of vertices
$V_n$ and the set of edges $2^{V_n} - \{ \emptyset , V_n \}$. If there
is a permutation $\sigma$ of $V_n$ and a set $E \subset 2^{V_n} - \{
\emptyset , V_n \}$ such that $\{ E, \sigma (E),..., \sigma^{p-1}(E)
\}$ is a $p$-partition of $ 2^{V_n} - \{ \emptyset , V_n \}$ then we
call it a cyclic $p$-partition of $K_n$ and we say that $\sigma$ is
$p$-complementing. We prove that $\tilde{K}_n$ has a cyclic
$p$-partition if and only if $p$ is prime and $n$ is a power of $p$
(and $n > p$). Moreover, any $p$-complementing permutation is cyclic.
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