Rook theory of the finite general linear group

Matrices over a finite field having fixed rank and restricted support are a natural qanalogue of rook placements on a board. We develop this q-rook theory by defining a corresponding analogue of the hit numbers. Using tools from coding theory, we show that these q-hit and q-rook numbers obey a variety of identities analogous to the classical case. We also explore connections to earlier q-analogues of rook theory, as well as settling a polynomiality conjecture and finding a counterexample of a positivity conjecture of the authors and Klein.