Combinatorics of diagrams of permutations

There are numerous combinatorial objects associated to a Grassmannian permutation wλwλ that index cells of the totally nonnegative Grassmannian. We study several of these objects and their q-analogues in the case of permutations w that are not necessarily Grassmannian. We give two main results: first, we show that certain acyclic orientations, rook placements avoiding a diagram of w, and fillings of a diagram of w are equinumerous for all permutations w. Second, we give a q-analogue of a result of Hultman–Linusson–Shareshian–Sjöstrand by showing that under a certain pattern condition the Poincaré polynomial for the Bruhat interval of w essentially counts invertible matrices over a finite field avoiding a diagram of w. In addition to our main results, we include at the end a number of open questions.