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arXiv · 1611.03387

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

Dylan Heuer·Chelsey Morrow·Ben Noteboom·Sara Solhjem·Jessica Striker·Corey Vorland
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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 $n\times n$ 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.

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Dylan Heuer, Chelsey Morrow, Ben Noteboom, Sara Solhjem, Jessica Striker, Corey Vorland. 2016-11-10. Chained permutations and alternating sign matrices - inspired by three-person chess. https://doi.org/10.1016/j.disc.2017.08.005

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