  $q$-Kaplansky numbers were considered by Chen and Rota. We find that
$q$-Kaplansky numbers are connected to the symmetric differences of Gaussian
polynomials introduced by Reiner and Stanton. Based on the work of Reiner and
Stanton, we establish the unimodality of $q$-Kaplansky numbers. We also show
that $q$-Kaplansky numbers are the generating functions for the inversion
number and the major index of two special kinds of $(0,1)$-sequences.
Furthermore, we show that $q$-Kaplansky numbers are strongly $q$-log-concave.
