We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We use machinery developed in the study of sign matrix polytopes to determine the inequality descriptions, facet enumerations, and face lattices of these polytopes. We also study partial permutohedra that we show arise naturally as projections of these polytopes. We directly prove vertex and facet enumerations and also characterize the face lattices of partial permutohedra in terms of chains in the Boolean lattice.
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.
We investigate analogues of alternating sign matrices, called partial alternating sign matrices. We prove bijections between these matrices and several other combinatorial objects. We use an analogue of Wieland's gyration on fully-packed loops, which we relate to the study of toggles and order ideals. Finally, we show that rowmotion on order ideals of a certain poset and gyration on partial fully-packed loop configurations are in equivariant bijection.
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