The problem of counting subspaces of a finite vector space according to their profiles with respect to a linear endomorphism was posed by Bender, Coley, Robbins, and Rumsey in 1992.  This enumeration problem includes, as a special case, a problem posed by H. Niederreiter in the context of pseudorandom number generation. In this talk, we present a complete solution to these counting problems by giving an explicit formula in terms of symmetric functions. The formula is expressed as a Hall scalar product involving dual \(q\)-Whittaker functions and another symmetric function that is determined by conjugacy class invariants of the linear endomorphism.

 As immediate consequences, we uncover  combinatorial finite field interpretations for coefficients in \(q\)-Whittaker expansions of several classical symmetric functions, including power sums, complete homogeneous and products of modified Hall-Littlewood polynomials. Additionally, we apply these results to count anti-invariant subspaces which were originally defined by Barría and Halmos, and discuss some connections with Krylov subspace theory. We also highlight a recent development connecting these enumeration problems to point counting on Hessenberg varieties over finite fields.