Simple operators and $q$-Whittaker coefficients of power sum symmetric functions

We give a new proof of a theorem of Bender, Coley, Robbins and Rumsey on counting subspaces with a given profile with respect to a simple operator. Counting such subspaces is equivalent to the problem of determining the $q$-Whittaker coefficients in the expansion of the power sum symmetric function. As a consequence we obtain a result of Chen and Tseng which answers a problem of Niederreiter on splitting subspaces.