Matrices with restricted entries over finite fields

We study the number of matrices of a given rank over a finite field with specified entries forced to be zero. We show that this number gives a q-analogue of counts of certain partial permutations or rook numbers. In particular, we consider the case of invertible matrices with zero diagonal, giving a q-analogue of derangement numbers. We also consider invertible symmetric and invertible skew-symmetric matrices with zero diagonal as well as discuss some related questions.

This talk is based on joint work with Joel Lewis, Alejandro Morales, Greta Panova, Steven Sam, and Yan Zhang.