Someone here was asking me a few days ago what was known about counting the number of different ways to recursively partition space by axis-aligned hyperplanes. Unfortunately I don't remember who it was, now that I've run across what looks like a very relevant looking reference: The number of guillotine partitions in d dimensions, Eyal Ackerman, Gill Barequet, Ron Y. Pinter, and Dan Romik, Inf. Proc. Lett. 98 (4): 162–167, 2006, doi:10.1016/j.ipl.2006.01.011. So in hopes of either that person running across this post, or being able to look it up here in case they ask me again, here it is.