Can someone please proof how number of permutations of first N natural numbers with no three-term increasing subsequence can be given by the Nth catalan number .. ?
More formally number of permutations such that there doesn't exist any i<j<k for which a[i]<a[j]<a[k].
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Catalan Number is the number of different ways a convex polygon with n + 2 sides can be cut into triangles by connecting vertices with straight lines (a form of Polygon triangulation).Try to find similarity between your problem and this problem. I do not have any formal proof as of now. Hope it helps.
https://math.stackexchange.com/questions/155867/penguin-brainteaser-321-avoiding-permutations
check this