Don't touch me

Let $a_k$ be the number of valid configurations for $k$ points. Pick an arbitrary point $p_1$ . The point $p_1$ can either not be used as the end point of a chord, in $a_{k-1}$ ways, or it can be connected to any other point $p_2$ , in which case it splits the remaining points into two groups without chords between them. Both smaller groups can be considered to be independent smaller instances of the same problem, so we can multiply their number of configurations.

This can be written as $a_k = a_{k-1} + \sum_{i=0}^{k-2} a_i \times a_{k-2-i}$ , which, starting with $a_0 = 1$ , gives the sequence:

$1, 1, 2, 4, 9, 21, 51, 127, 323, 835, \ldots$

Let $S_k$ be the number of ways to draw chords as described above with points labelled $1,2, \ldots k.$ Then $S_0 = S_1 = 1,$ and we see that $S_2 = 2$ since we can either draw the chord between the points or not.

We claim that $S_k = S_{k-1} + \sum\limits_{i=0}^{k-2} S_{i}S_{k-2-i}.$ First note that there are $S_{k-1}$ ways to draw chords such that no chord uses the point $k.$ The number of ways to draw the chords so that there is a chord from $k$ to $j$ is $S_{j-1}S_{k-j-1},$ since this chord divides the remaining points into two sets of size $j-1$ and $k-j-1$ respectively. As $j$ ranges from $1$ to $k-1$ we get the claimed sum.

Using the recursive formula $S_k = S_{k-1} + \sum\limits_{i=0}^{k-2} S_{i}S_{k-2-i},$ we construct the following table of values for $S_{k}.$

$\begin{array}{|r|cccccccccc|} \hline k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ \hline S_k & 1 & 1 & 2 & 4 & 9 & 21 & 51 & 127 & 323 & 835\\ \hline \end{array}$

Thus, there are $835$ ways to choose a set of chords such that none of them intersect.

The number of ways of drawing non-intersecting chords between $k$ points is given by the Motzkin number $M_k$ .

As such the number of non-intersecting chords which we can draw between $9$ points is the $9th$ Motzkin number which is given by $M_9$ , which is equal to $\fbox{835}$ .