MathDB

the function $f^{n-2}$ is constant, but $f^{n-3}$ is not

Source: Moldova TST 1996

August 8, 2023

function

Problem Statement

Let

A{}

be a set with

n{}

(n\geq3)

elements. Iterations

f^2,f^2,\ldots

of the function

f:A\rightarrow A

are defined as

f^2(x)=f(f(x)), f^{i+1}=f(f^i(x)), \forall i\geq2

. Find the number of functions

f:A\rightarrow A

with the property: the function

f^{n-2}

is constant, but

f^{n-3}

is not.