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**Shanks** A Challenge Problem:

Let $\displaystyle S_n$ be all the permutations of {1,2,...,n}.

for any given permutation $\displaystyle \sigma$, if $\displaystyle \sigma(i)=i$, we say $\displaystyle i$ is a fix point of $\displaystyle \sigma$.

A permutation may have several fix points, let $\displaystyle FP(\sigma)$ be the set of fix points of the permutation $\displaystyle \sigma$.

Problem: How many permutations are there whose fix-points set has exactly k elements(points)? k=0,1,...,n.