How many unique ways to repeatedly cut a rectangle? I am currently reading this book ("Plans de métro et réseaux neuronaux - La théorie des graphes" by Alsina Claudi) where the author presents this open problem without naming it and I'd like to find out more about it.
The problem is as follows: 
Starting from a rectangle you may cut it horizontally or vertically, and recursively do so on any of the sub-rectangles. How many unique ways are there to perform $N$ cuts?
The author claims that for $n=0, 1, 2, 3, 4,$ and $5$ cuts there are respectively $1, 1, 2, 7, 22,$ and $117$ unique subdivisions, however for $n>5$ the number of unique subdivisions is unknown.
I am intrigued as I cannot find the above sequence on the OEIS website.
Page of the book with helpful visuals (in French).
Could anyone point me to where I could find out more about this open problem please?
 A: There are several phrasings that are fruitful when looking for research papers about this problem. What we're looking for is called either a "rectangular dissection"/"dissection into rectangles" or a "rectangular partition"/"partition into rectangles" of a square. I have found several papers on related problems, though none that are exactly identical to this one.
For instance, there is Dissecting a square into rectangles of equal area by Häggkvist, Lindberg, and Lindström. The equal-area restriction is actually not a huge difference: given a particular topology of a rectangle dissection, we can make the rectangles have equal area. To do this, simply make sure that when you make a cut that's supposed to have $k$ pieces on one side and $\ell$ on the other at the end, that the cut divides its sub-rectangle in a $k:\ell$ ratio.
There are two ways this problem is still different from the one posed in the French textbook, however:

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*The equal-area condition forces a dissection like the one in Figure 1, below, to look like Figure 2 instead. So pictures like the one in Figure 1 are missed.

*There are rectangular dissections that are not obtained by repeatedly dividing a single sub-rectangle into two pieces: for example, the dissection in Figure 3  has $5$ small rectangles, no proper subset of which forms a bigger rectangle.

Along the same lines, there's Rectangular dissections of a square by Boros and Füredi which generalizes the problem to dissections into $n$ rectangles with prescribed areas $w_1, w_2, \dots, w_n$ such that $w_1 + w_2 + \dots + w_n = 1$. This paper gives quite precise asymptotic bounds, which may be interesting.

If you want to avoid pictures like Figure 3, there is a term for this as well. A cut is called a guillotine cut if it breaks a connected area into two parts, and a rectangular partition is a guillotine rectangular partition if it can be formed by a sequence of guillotine cuts. Unfortunately, the papers I have found here mostly investigate algorithms for how to optimally (or near-optimally) partition a given complicated shape into rectangles, and aren't about the square.
