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

Abstract

Abstract. 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.

Partial Permutation and Alternating Sign Matrix Polytopes

“…We allow these matrices to be rectangular, and in doing so relax the condition that each row and column must sum to 1. These matrices appear as the individual components of chained alternating sign matrices (see [12]), which was the original motivation for studying them. We hope this study of partial alternating sign matrices will prove useful for future research in the chained setting.…”