Chained permutations and alternating sign matrices-Inspired by three-person chess

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×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.