MathDB

IOQM 2022-23 P-20

Source:

October 30, 2022

combinatoricsLandmark point

Problem Statement

For an integer

n\ge 3

and a permutation

\sigma=(p_{1},p_{2},\cdots ,p_{n})

of

\{1,2,\cdots , n\}

, we say

p_{l}

is a

landmark

point if

2\le l\le n-1

and

(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0

. For example , for

n=7

,\\ the permutation

(2,7,6,4,5,1,3)

has four landmark points:

p_{2}=7

,

p_{4}=4

,

p_{5}=5

and

p_{6}=1

. For a given

n\ge 3

, let

L(n)

denote the number of permutations of

\{1,2,\cdots ,n\}

with exactly one landmark point. Find the maximum

n\ge 3

for which

L(n)

is a perfect square.

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