MathDB

A function

f:\{ 1,2,3,\cdots ,2016\}\rightarrow \{ 1,2,3,\cdots , 2016\}

is called good if the function

g(n)=|f(n)-n|

is injective. Furthermore, a good function

f

is called excellent if there exists another good function

f'

such that

f(n)-f'(n)

is nonzero for exactly one value of

n

. Let

N

be the number of good functions that are not excellent. Find the remainder when

N

is divided by

1000

.
Proposed by Nathan Ramesh

Problem Statement