Matrices with restricted entries and q-analogues of permutations

TL;DR

This paper explores matrices over finite fields with restricted entries, establishing connections to q-analogues of permutations, deriving formulas for special matrix counts, and linking these to Lie theory and polynomial enumeration.

Contribution

It introduces new formulas and recursions for counting matrices with zero diagonal and restricted entries, framing these in the context of q-analogues and Lie theory.

Findings

Closed formula for invertible matrices with zero diagonal

q-analogue of derangements

Relationships between skew-symmetric and symmetric matrices

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, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.