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

TL;DR

This paper introduces new permutation families based on rook placements on chained chessboards, explores their enumeration, and establishes connections to alternating sign matrices and related combinatorial objects.

Contribution

It defines and enumerates chained permutation families and chained alternating sign matrices, providing bijections to various combinatorial structures and extending classical concepts.

Findings

Enumeration formulas for chained rook placements

Bijections to matrix forms and graph matchings

Enumeration of chained alternating sign matrices for specific parameters

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.