How many onto functions can exist between two finite sets?

Asked Nov 25 '16 at 17:45

Active Nov 25 '16 at 19:37

Viewed 67 times

How can I approach this problem ? I tried to use combinations and permutations but I can't understand how to apply combinatorics .

I tried many things in vain . Please introduce new concepts to me if required .

edited Nov 25 '16 at 19:37

ziggurism

asked Nov 25 '16 at 17:45

InquisitiveMind

If $A$ and $B$ are two finite sets and if $|A|>|B|$, then there is no onto function from $B$ to $A$. You need to specify about cardinality of the sets. – Nov 25 '16 at 17:48
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See Number of onto maps. – StubbornAtom Nov 25 '16 at 17:48
Thanks but in the other case ? – InquisitiveMind Nov 25 '16 at 17:50
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Did you mean "How many" rather than "How any" in the question title? – David K Nov 25 '16 at 17:57
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Let $A={1,2,...,k}$ and $B={1,2,...,n}$ and for simplicity write the mapping $f:A\to B$ as $f:[k]\to [n]$.
- If $n>k$, there are no surjections.
- If $n=k$, every surjection is a bijection and the number of maps is $n!$
- If $n<k$, number of surjections is given by the Inclusion-exclusion principle: $\sum_{i=0}^{n-1}(-1)^i\binom{n}{i}(n-i)^k$.
– StubbornAtom Nov 25 '16 at 18:01
@StubbornAtom In fact the formula $\sum_{i=0}^{n-1}(-1)^i\binom{n}{i}(n-i)^k$ is correct for all $k\gt0$ and even for $k=0$ if you rewrite it as $\sum_{i=0}^{n}(-1)^i\binom{n}{i}(n-i)^k.$ – bof Nov 25 '16 at 20:21
@David K I meant " How many " in the question title – InquisitiveMind Nov 26 '16 at 15:34

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