There aren't patterns with the following exceptions:
1) you are more likely to get artifacts that you don't already have, which is favorable for completing sets.
2) some artifacts are restricted to a particular area, such as the Frozen artifacts which drop in the Frozen area
3) the holiday artifacts only drop during the relevant "holiday" period.
In my experience, once you start completing sets, it tends to go more quickly. That is, once you have all the rings, rings appear less often and you are more likely to complete the other sets you don't already have. Similarly, with the holiday ones, those tend to turn up steadily until you complete them (and get the hat) and then you seem them only rarely after that.