Let 2n (equally spaced) points on a circle be chosen. Show that the number of ways to join these points in pairs, so that the resulting n line segments do not intersect, equals the nth Catalan number C_n.
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There are 2n points on a circle. We want to divide them into pairs and connect each pair with a segment (i.e. a chord) in such a way that these segments do not intersect. Show that the number of ways to do this is given by a Catalan number.
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Show that the Catalan numbers count the following. (a) Sequences x1x2 · · · xn of positive integers such that x1 ≤ x2 ≤ · · · ≤ xn and for each i, xi ≤ i. For example, for n = 3, the sequences counted are 111, 112, 113, 122, 123. (b) Ways to join 2n points on a circle with n non-intersecting chords. For example, for n = 3, the ways counted are
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