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February 24th, 2009, 06:20 AM   #1
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Complexity of Problem

Hi,

I have the following combinatorial problem.

Let P and Q be two sets and let M: P -/-> Q be a partial and injective mapping from P to Q. What is the function that describes the number of allowed mappings from P to Q?

For example, if I have the set {a,b} and the set {A,B,C} then the following mappings are allowed:
a -> A, b -> B
a -> A, b -> C
a -> A, b ->
a -> B, b -> A
a -> B, b -> C
a -> B, b ->
...
In this case the number of allowed mappings is 13, but what is the general function?

I need this function to give an indication of the complexity of a problem. I would say that the problem has a lower bound of O(N!).

Any help would be appreciated.
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February 24th, 2009, 07:25 AM   #2
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Re: Complexity of Problem

This is Sloane's A088699.

Let |P| = p and |Q| = q. Then mappings(p, q) = sum(k=0, min(p, q), binomial(p, k) * binomial(q, k) * k!). The function is symmetric in p and q.

If |P| = p = |Q|, then log(mappings(p)) ~ p log (p/e) + 2*sqrt(p) + log(p)/4.
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August 26th, 2009, 08:05 AM   #3
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Re: Complexity of Problem

I finally got a chance to look take a look at your answer. It seems to be exactly what I need. Thanks a lot.
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