Question 74
Let $w = w_{1}, w_{2}, \ldots, w_{6}$ be a permutation of the integers $\{1, 2, \ldots, 6\}$. If there do not exist indices $i \lt j \lt k$ such that $w_{i} \lt w_{j} \lt w_{k}$ or indices $i \lt j \lt k \lt l$ such that $w_{i} \gt w_{j} \gt w_{k} \gt w_{l}$, then $w$ is said to be exquisite. Find the number of exquisite permutations.