Carpet on stringers is normal when the stringer wood isn't a finished product (2X12), the carpet is held on by staples (1/4 X 9/16 staples). The carpet is cut down from the side of the carpet, EX: if your carpet is a 12 X 7 piece you will get twelve 1' X 7' pieces. Start on the top of the stringer and staple the carpet closest to the wall all the way down the piece of carpet, carefully cut the carpet in down to each step and staple to secure as you go. Continue down the stringer using the same process.
This is not as popular any more, but some woods aren't suitable for paint or stain.