# What is the number of ways to divide a rectangle into $n$ smaller rectangles line by line? [duplicate]

The original problem was to consider how many ways to make a wiring diagram out of $n$ resistors. When I thought about this I realized that if you can only connect in series and shunt. - Then this is the same as dividing an area with $n-1$ horizontal and vertical lines. When each line only divides one of the current area sections into two smaller ones.

This is also the same as the number of ways to make a set of $n$ (and only $n$) rectangles into a bigger rectangle. If the rectangles can be drawn by dividing the big rectangle, line by line, into the set of rectangles without lose endpoints of the line. - Can someone come to think of "a expression of $n$" which equals this amount, independent of the order of the rectangles or position?

(It is only the relations between the area sections that matters and not left or right, up or down. However dividing an area with a horizontal line is not the same as dividing it with a vertical line.)

• Not all subdivisions of a rectangle are of this kind. For instance in a $3\times3$ square you can single out the central square and partition the remaining $8$ squares into $4$ dominos. Apr 24, 2013 at 9:13
• Thanks, I notised that recently and will edit the title again... Apr 24, 2013 at 14:48
• A useful approach to problems like this is to calculate small values by hand and search oeis.org That often has nice answers and references. The symmetries make this one difficult, even for n=4, though. Apr 24, 2013 at 15:30
• I believe this is the serie: A000084 but I'm not sure, can you check it? Apr 24, 2013 at 22:12
• Here is an illustration of the problem where $f(n)$ is the function we seek: Link Apr 25, 2013 at 19:48

For the original problem, which I understand as dividing a rectangle into $n$ rectangles where each straight cut splits one piece in two, the answer is $n - 1$, as is easy to prove by (strong) induction.