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{\bf Nigel Martin and Richard Stong}
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{\bf Permutation Separations and Complete Bipartite Factorisations of $K_{n,n}$}
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Suppose $p<q$ are odd and relatively prime. In this paper we complete
the proof that $K_{n,n}$ has a factorisation into factors $F$ whose
components are copies of $K_{p,q}$ if and only if $n$ is a multiple of
$pq(p+q)$. The final step is to solve the ``c-value problem'' of
Martin. This is accomplished by proving the following fact and some
variants: For any $0\le k\le n$, there exists a sequence
$(\pi_1,\pi_2, \dots,\pi_{2k+1})$ of (not necessarily distinct)
permutations of $\{1,2,\dots,n\}$ such that each value in
$\{-k,1-k,\dots,k\}$ occurs exactly $n$ times as $\pi_j(i)-i$ for
$1\le j\le 2k-1$ and $1\le i\le n$.

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