For an integer n 3 and a permutation = (p₁, p₂, , p_n ) of 1,2, , n , we say p₁ is a landmark point if 2 I n-1 and (p_ I-1 -p_I ) (p_ I+1 -p_I )>0 . For example, for n=7 , the permutation (2,7,6,4,5,1,3) has four landmark points: p₂=7, p₄=4, p₅=5 and p₆=1 . For a given n 3 , let L(n) denote the number of permutation of 1,2, , n with exactly only landmark point. Find the maximum n 3 for which L(n) is a perfect square.

Step-by-step solution

For the permutations of set 1,2,3, , n , the landmark point should be 1 or n to satisfy given conditions. I ^ st II ^ nd - r ^ th - - n ^ th If n is at (similarly for 1 ) r^ th position, there is only one permutation of the remaining numbers for each selection. aligned So, number of selections & = _ r=2 ^ n-1 n-1 r-1 & =2^ n-1 -2 aligned Total number of selections =2 (2^ n-1 -2 )L(n)=4 (2^ n-2 -1 ) Now for L(n) to be a perfect square n should be equal to 3 . n=3