Concept explainers
Let 2n(equally spaced) points on a circle be chosen. Show that the number ofways to join these points in pairs, so that the resulting nline segments do notintersect, equals the nth Catalan number Cn.
To show: That the number of ways to join the 2n (equally spaced) points in pairs.
Explanation of Solution
The number of ways to join the 2n (equally spaced) points in pairs, so that the resulting n line segments do not intersect, equals the nth Catalan number
The
Let
For a matching in M let t denote the point matched to point 1. Note that t is even. For
The sets
For
Therefore
By these comments
Where,
Consider the generating function
Using the Catalan and obtain
Using the quadratic formula
In other words
Using Newton’s binomial theorem this becomes
For this equation at
Consequently, for
Note that,
Hence, proved.
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