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 \e 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.