  We discuss the problem posed by Bender, Coley, Robbins and Rumsey of
enumerating the number of subspaces which have a given profile with respect to
a linear operator over the finite field $\mathbb{F}_q$. We solve this problem
in the case where the operator is diagonalizable. The solution leads us to a
new class of polynomials $b_{\mu\nu}(q)$ indexed by pairs of integer
partitions. These polynomials have several interesting specializations and can
be expressed as positive sums over semistandard tableaux. We present a new
correspondence between set partitions and semistandard tableaux. A close
analysis of this correspondence reveals the existence of several new set
partition statistics which generate the polynomials $b_{\mu\nu}(q)$; each such
statistic arises from a Mahonian statistic on multiset permutations. The
polynomials $b_{\mu\nu}(q)$ are also given a description in terms of
coefficients in the monomial expansion of $q$-Whittaker symmetric functions
which are specializations of Macdonald polynomials. We express the
Touchard--Riordan generating polynomial for chord diagrams by number of
crossings in terms of $q$-Whittaker functions. We also introduce a class of
$q$-Stirling numbers defined in terms of the polynomials $b_{\mu\nu}(q)$ and
present connections with $q$-rook theory in the spirit of Garsia and Remmel.
