LAUSR

ABSTRACT Matrices over a finite field having fixed rank and restricted support are a natural q-analog of rook placements on a board, even though their enumeration is known to yield nonpolynomial answers in some cases. We develop this q-rook theory by defining a corresponding analog 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-analogs of rook theory, as well as settling a polynomiality conjecture and finding experimentally a counterexample of a positivity conjecture of the authors and Klein.