How many into and onto functions are there from [k] to [n]?
a) If k > n? Thus far I have S(k,n)n! However, I don't know how to calculate that.
b) If k <= n?
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For into functions we need $k\leq n$ and if so there are $\frac{n!}{(n-k)!}$ since there are $n$ options for the image of 1, then $n-1$ options for the image of $2$ and so on untill $n+1-k$ options for the image of $k$. So the answer is $n\cdot(n-1)\cdot \dots \cdot (n+1-k)=\frac{n!}{(n-k)!}$
for the onto part we want to partition the codomain into $k$ parts and then assign one each of those parts to one of the elements of the domain. Thus there are ${n \brace k}k!$
Asinomás
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$n \brace k$ is the notation I use for the stirling numbers of the second kind. – Asinomás Dec 06 '14 at 02:55