  We define and enumerate two new two-parameter permutation families, namely,
placements of a maximum number of non-attacking rooks on $k$ chained-together
$n\times n$ chessboards, in either a circular or linear configuration. The
linear case with $k=1$ corresponds to standard permutations of $n$, and the
circular case with $n=4$ and $k=6$ corresponds to a three-person chessboard. We
give bijections of these rook placements to matrix form, one-line notation, and
matchings on certain graphs. Finally, we define chained linear and circular
alternating sign matrices, enumerate them for certain values of $n$ and $k$,
and give bijections to analogues of monotone triangles, square ice
configurations, and fully-packed loop configurations.
