  There are numerous combinatorial objects associated to a Grassmannian
permutation $w_\lambda$ 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-Sj\"ostrand by showing that under a certain pattern
condition the Poincar\'e polynomial for the Bruhat interval of $w$ essentially
counts invertible matrices avoiding a diagram of $w$ over a finite field. In
addition to our main results, we include at the end a number of open questions.
