Say I have a long rectangular hole of size n and then a bunch of smaller rectangles of different lengths a, b, c, etc such that the sum of their lengths is <=n.
I am trying to figure out the number of distinct permutations. For instance, if I had a hole of length 4 and two rectangles of length 2, there's only one way to arrange that. But if I had one of length 3 and one of length 1, that's two ways to arrange it.
If I had a hole of length 4 and a sub-rectangle of length 2, there are 3 ways, and so on.
Edit for clarification: The lengths are integers only, and they can only be "snapped" in at integer intervals (so you can't place a rectangle at, say, 1.233249 and then claim there are infinitely many ways to insert a rectangle). Area also doesn't matter... the height of the rectangles are all the same. It's only the lengths that differ.
Is there a simple mathematical combinatoric that can be applied here?