Permutations and directions of Integers

[chaotixmonjuish]

Given this permutations {1,2...,n}, prove that the directions of 1 and 2 never change.

Proof: When generating permutations, one starts with everything having a left facing arrow. In order to determine what is mobile, the arrow must be pointing towards a smaller integer. 1 points to nothing so it is no mobile. 2 is mobile, however once it moves to the left it is no longer mobile. The remainder of the integers in {1,2,...,n} are mobile. Therefore, 1 and 2 will always point to the left since all the other integers are larger than those two.

 

[Dick]

I don't see any arrows here. I don't know what your problem or description means. Can you explain using a more standard mathematical notation?

 

[chaotixmonjuish]

I don't know how to put arrows above the numbers.

 

[Dick]

That's ok. I wouldn't understand what it meant anyway. What does the problem mean? How is {1,2,3,...,n} a permutation? Do you mean the cycle (1,2,3,...,n)? Your notation and adjectives like 'mobile' and 'direction' are pretty obscure for me.

 

[chaotixmonjuish]

Basically you want to track how integers move in a permutation. I'm not sure how to explain it more than that. And I need to prove that 1 and 2 will always have a left facing direction.