Non-attacking rook placements on crossword grids

Source abstract

We introduce the notion of a non-attacking rook placement on a crossword grid. A crossword grid is a collection of white squares (which comprise across and down words) and black squares (which separate the words), and a complete non-attacking rook placement on such a grid is a subset of white squares which intersects every across and every down word exactly once. We prove an upper bound on the number of rook placements that a general grid can admit. We then study sparse grids in which no two black squares share an edge and show that rook placements on certain sparse grids correspond bijectively to alternating sign matrices with prescribed $-1$ entries. Specializing further to permutation grids, we prove that every permutation grid admits at least one rook placement, and characterize the permutations whose grids admit exactly one placement in terms of the Robinson--Schensted correspondence. Throughout, we pose a variety of conjectures and open questions.