A permutation of is a functional graph where f(i) = p[i]. Each element points to its image under the permutation. This is a direct correspondence. Every permutation decomposes into disjoint cycles. Following any element repeatedly leads back to itself.
The cycle structure determines permutation properties. Finding these cycles uses the same techniques: DFS, Floyd's algorithm, or simulation. Permutation problems often reduce to functional graph problems. The techniques transfer directly.