A generic rectangulation is a tiling of a rectangle by rectangles, with no four rectangles sharing a single corner (think: no Arizona, Colorado, New Mexico and Utah). For example:

We want to ignore lengths of edges and just look at the different configurations of rectangles. In this way of thinking, the rectangulations and are the same, but both are different from . So how many different rectangulations are there with n rectangles? The answers, for n from 1 to 5, are 1, 2, 6, 24 and 116, and I can compute the answers for n up to 27. But what’s the formula? I don’t know yet, but surprisingly (or not), the answer is connected to pattern-avoiding permutations. This talk will be accessible to all undergraduates. I won’t assume any prior knowledge of rectangulations or permutations.