The Eulerian generating function of q-derangements

Let FqFq denote the finite field with q   elements. For nonnegative integers n,kn,k, let dq(n,k)dq(n,k) denote the number of n×nn×nFqFq-matrices having k   as the sum of the dimensions of the eigenspaces (of the eigenvalues lying in FqFq). Let dq(n)=dq(n,0)dq(n)=dq(n,0), i.e., dq(n)dq(n) denotes the number of n×nn×nFqFq-matrices having no eigenvalues in FqFq. The Eulerian generating function of dq(n)dq(n) has been well studied in the last 20 years [Kung, The cycle structure of a linear transformation over a finite field, Linear Algebra Appl. 36 (1981) 141–155, Neumann and Praeger, Derangements and eigenvalue-free elements in finite classical groups, J. London Math. Soc. (2) 58 (1998) 564–586 and Stong, Some asymptotic results on finite vector spaces, Adv. Appl. Math. 9(2) (1988) 167–199]. The main tools have been the rational canonical form, nilpotent matrices, and a q-series identity of Euler. In this paper we take an elementary approach to this problem, based on Möbius inversion, and find the following bivariate generating function:∑n,k⩾0dq(n,k)ykxnqn2(n)!=∑n⩾0(y-1)(y-q)⋯(y-qn-1)xnqn2(n)!q∑n⩾0qn2xnqn2(n)!.