I'd call that a saddle clamp, but the Brompton term appears to be 'Pentaclip'. A seat clamp is probably something even more proprietary and squeaky, that you'd find on a recumbent, but it sounds a lot like 'seatpost clamp', which is one of the Brompton's known quirky components, which is why we all assumed you were talking about that.
The Brompton one is clever, in that it can be twiddled round to adjust the height. No idea about the diameter of the post and compatibility thereof, but it certainly fits standard saddles.