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Permutations with order dividing twelve

High school · ●●○○○ 20 pts

Let A={1,2,…,6}A=\{1,2,\dots,6\} and let f:A→Af: A \to A be a one-to-one mapping. Define f(2)(x)=f(f(x))f^{(2)}(x)=f(f(x)) and f(n+1)(x)=f(f(n)(x))f^{(n+1)}(x)=f(f^{(n)}(x)) for n≥1n \ge 1. Find the number of mappings ff such that f(12)(x)=xf^{(12)}(x)=x for every x∈Ax \in A.

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permutationscycle decompositionfunctional iterationcounting