Rook theory of the finite general linear group

Abstract

Matrices over a finite field having fixed rank and restricted support are a natural $q$-analogue 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.

25 pages, 10 figure files. Minor change in definition of q-hit numbers changes notation but doesn't substantively affect results

References in corpus (2)

• Integer sequences and matrices over finite fields
• Combinatorics of diagrams of permutations