There can only be one

Discrete Mathematics Level pending

A unimodal permutation is a permutation with only one local maximum. That is, a unimodal permutation of \(n\) elements, \(\sigma_1,\sigma_2,\cdots,\sigma_n\), must have \(\sigma_1 < \sigma_2 < \cdots < \sigma_k\) and \(\sigma_k > \sigma_{k+1} > \cdots > \sigma_n\) for some positive integer \(k \le n\).

How many unimodal permutations are there of the set \(\{1,2,3,4,5,6,7,8,9\}\)?

(Adapted from Analytic Combinatorics by Philippe Flajolet)

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