TITLE: Number of Ways of Partitioning a Rectangle
QUESTION [5 upvotes]: Given a rectangle of integer sidelengths $m\times n$, consider partitioning it into smaller rectangles also of integer sidelengths. How many such partitions are possible?
I wonder if this is just an exercise-level problem or something that hasn't been solved... it is such a simple question that was probably studied before.

REPLY [2 votes]: There is a nice discussion of a more simple problem here: Number of ways to partition a rectangle into n sub-rectangles Given its complexity, I have doubts that your problem has a nice closed-formed answer.