grandes-ecoles 2019 Q37

grandes-ecoles · France · centrale-maths2__pc Permutations & Arrangements Probability via Permutation Counting

For every integer $n \geqslant 2$, let $p_n$ be the probability that a uniformly random permutation of $\llbracket 1, n \rrbracket$ is alternating up (with $p_0 = p_1 = 1$). Show that the sequence $(p_n)$ tends to 0. Give an equivalent of $p_{2n+1}$ as $n$ tends to infinity.

For every integer $n \geqslant 2$, let $p_n$ be the probability that a uniformly random permutation of $\llbracket 1, n \rrbracket$ is alternating up (with $p_0 = p_1 = 1$). Show that the sequence $(p_n)$ tends to 0. Give an equivalent of $p_{2n+1}$ as $n$ tends to infinity.

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