For an integer n ≥ 3 and a permutation σ=(p_1, p_2, , p_n) of \...

For the permutations of set $\{1,2,3, \ldots, n\}$, the landmark point should be 1 or $n$ to satisfy given conditions. \overline{\mathrm{I}^{\mathrm{st}}} \overline{\mathrm{II}^{\mathrm{nd}}}-\overline{\mathrm{r}^{\mathrm{th}}}-\cdots-\overline{\mathrm{n}^{\mathrm{th}}} If $n$ is at (similarly for 1 ) $r^{\text {th }}$ position, there is only one permutation of the remaining numbers for each selection. $$\begin{aligned} \text { So, number of selections } & =\sum_{r=2}^{n-1}{ }^{n-1} C_{r-1} \\ & =2^{n-1}-2 \end{aligned}$$ $\therefore \quad$ Total number of selections $=2\left(2^{n-1}-2\right)$L(n)=4\left(2^{n-2}-1\right)$Now for$L(n)$to be a perfect square$n$should be equal to 3 .$\therefore \quad n=3