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Find all natural numbers n such that p_n > \frac{1}{3}

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October 30, 2005
probabilityinequalitiesfunctioncombinatorics unsolvedcombinatorics

Problem Statement

From the set of all permutations ff of {1,2,...,n}\{1, 2, ... , n\} that satisfy the condition: f(i)≥i−1f(i) \geq i-1 i=1,...,ni=1,...,n one is chosen uniformly at random. Let pnp_n be the probability that the chosen permutation ff satisfies f(i)≤i+1f(i) \leq i+1 i=1,...,ni=1,...,n Find all natural numbers nn such that pn>13p_n > \frac{1}{3}.
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