Western: sit more on his rump. A chair seat, at least a modified one, helps. So does a long leg, which seems the opposite, but shorter stirrups often result in gripping with the knee and creating a fulcrum point that the rump can bounce around.
Dave Carter, old time Spur cowpuncher, riding "Butterbean" while the herd of the last roundup on the old Espuela (Spur) Land and Cattle Company stop at a watering place. SMS Ranch, Texas., 1910
Pink Murray, famous wagon boss of the OR ranch in southern Arizona, mounted on his favorite horse. Navajo Indian blankets were used as saddle blankets, and "Pink" had a good one. OR Range, Arizona, 1909
English: use the thigh more and the seat bones less. The best advice I got was as soon as I get in the saddle, pull on the front of the saddle with the goal of forcing my knees as low as they will go. That seems to set me up to ride with a light rump but one that still doesn't bounce. I'm not a real English rider, but I do it sometimes...so take that advice for what you paid for it!