2005 OIM Problems/Problem 6

Problem

Given a positive integer $n$ , $2n$ points are aligned in a plane as $A_1, A_2,\cdots, A_{2n}$ . Each point is colored blue or red using the following procedure: In the plane, $n$ circles with end diameters $A_i$ and $A_j$ are drawn, disjoint two by two. Each $A_k$ , $1 \le k \le 2n$ , belongs to exactly one circle. The dots are colored so that the two points of the same circle have the same color. Find how many different colorations of the $2n$ points can be obtained by varying the $n$ circumferences and the distribution of colors.