The $k$-arrangements are permutations whose fixed points are $k$-colored. We prove enumerative results related to statistics and patterns on $k$-arrangements, confirming several conjectures by Blitvić and Steingrímsson. In particular, one of their conjectures regarding the equidistribution of the number of descents over the derangement form and the permutation form of $k$-arrangements is strengthened in two interesting ways. Moreover, as one application of the so-called decrease value theorem, we calculate the generating function for a symmetric pair of Eulerian statistics over permutations arising in our study. This generating function is expressed in terms of a newly introduced linear operator $\rho$ on formal power series.