i have absolutely zero math experience so i don't know if i'm using terms correctly but i was doing coding for a bot i'm making and got caught by a thinky bug so now i have to puzzle this out and would love if someone smarter than me could help with an answer.
basically, you have any number of elements which are attached together. elements can be attached by one-way chains, so, for example, a is "on top of" b, or b is "on top of" a, or a and b are on the same elevation, is the three possible pairings of a two-element group.

doing the math for three elements gives me 13 possible arrangements, but i have to do that by hand and i'd rather not do that for n = 4 or higher.

i hope i've explained this well enough, this is really making me think because there's clearly A pattern i'm just too dumb to figure it out.