A q-analogue of permutations are invertible matrices over a finite field with q elements. Indeed the number of such matrices is a natural q-analogue of n!. In this talk we consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. We show that these matrices are a q-analogue of permutations with restricted positions (i.e., placements of non-attacking rooks), and give a simple closed formula for the number of invertible matrices with zero diagonal, a q-analogue of the number of derangements. In addition, we study the question of when the number of matrices with given size, rank, and prescribed entries equal to 0 is a polynomial in q. For instance we generalize a result of Haglund by showing that when the support of the matrix lies in a skew shape, the number of such matrices is a polynomial with nonnegative coefficients. We also study the situation in which the prescribed 0s are the entries of the Rothe diagram of a permutation, and give intriguing conjectures in this case. This work is joint with Aaron Klein, Joel Lewis, Ricky Liu, Greta Panova, Steven Sam and Yan Zhang.

A derangement is a permutation with no fixed points. The number of such permutations has been studied even before Euler. Derangements are a special case of permutations with restricted positions also known as rook placements. Counting rook placements arises in areas like algebraic combinatorics, number theory, and statistical physics. We will present background and ideas on counting permutations with restricted positions. We will focus on families of restricted positions that have interesting properties. We will then generalize counting these permutations by looking at "q-analogues" of rook placements like the Garsia-Remmel q-rook numbers and invertible matrices over a finite field with restricted positions.