MathDB

For a positive integer

n

, consider all nonincreasing functions f : \{1,\hdots,n\}\to\{1,\hdots,n\}. Some of them have a fixed point (i.e. a

c

such that

f(c) = c

), some do not. Determine the difference between the sizes of the two sets of functions.
Remark. A function

f

is nonincreasing if

f(x) \geq f(y)

holds for all

x \leq y

Problem Statement