Find all possible $\{ x_1,x_2,...x_n \}$ permutations of $ \{1,2,...,n \}$ so that  when $1\le i \le n-2 $ then we have  $x_i < x_{i+2}$ and when $1 \le i \le n-3$ then we have  $x_i < x_{i+3}$ . Here $n \ge 4$.
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   - Therefore, the number of such permutations \( p_n \) is given by:
     \[
     p_n = F_{n+1}
     \]
   - This works for any \( n \geq 1 \).

The final answer is \( \boxed{ p_n = F_{n+1} } \).