February 24th, 2009, 06:20 AM  #1 
Newbie Joined: Feb 2009 Posts: 2 Thanks: 0  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. 
February 24th, 2009, 07:25 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  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. 
August 26th, 2009, 08:05 AM  #3 
Newbie Joined: Feb 2009 Posts: 2 Thanks: 0  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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