Let $n \in \mathbb{N}^{*}$. We are given $2 n$ points on the boundary of a circle. We denote $F_{n}$ as the number of ways to connect these points, two by two, using $n$ chords that do not intersect inside the circle. Find a recurrence relation between the $F_{n}$.
Let's choose an arbitrary point, for example, the one at the top of the circle. We partition the configurations according to the point to which the first one is connected. The concerned chord separates the configuration into two.

![](https://cdn.mathpix.com/cropped/2024_05_10_8c44cd8b5c585c15e58bg-431.jpg?height=345&width=332&top_left_y=1561&top_left_x=702)

Let $k$ be half the number of points strictly to the right of this chord. $k$ thus varies from 0 to $n-1$. The $2k$ points to the right of the chord can be connected in $F_{k}$ ways, and those to the left in $F_{n-k-1}$ ways. Therefore, we have

$$
F_{n}=\sum_{k=0}^{n-1} F_{k} F_{n-k-1}
$$