Authors: Joel Brewster Lewis ¹; Ricky Ini Liu ²; Alejandro H. Morales ¹; Greta Panova ³; Steven V Sam ¹; Yan Zhang ¹

• 1 Department of Mathematics [MIT]
• 2 School of Mathematics
• 3 Department of Mathematics [Cambridge]

We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.