Matrices with restricted entries and q-analogues of permutations (extended abstract)

Abstract

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.

Keywords

• linear algebra over finite fields
• $q$-analogues
• derangements
• [math.math-co] mathematics [math]/combinatorics [math.co]
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]