Let S = {1; 2; 3; 4; 5; 6; 7}

Analytically determine the number of maps f from S to S such that f²⁰⁸⁰(x) = x for every x belonging to S.

Note: The iteration is denoted by the superscript, such that f¹(x) = f(x) and
fⁿ(x) = f(f^n-1(x)) for all n > 1.