combinatorics

Suppose you are given a list L of n numbers and an integer k<n . Is there an efficient way to calculate the sum of all products of k distinct numbers in L ? As an example, take L=[1,3,4,6] and k=2 . Then the number I am looking for is 1*3 + 1*4 + 1*6 + 3*4 + 3*6 + 4*6 . Can you think of a way of doing it which avoids generating all the subsets of size k ? Let F(X,k,n) be the k-product sum of first n elements of array X. F(X,k,n) = F(X,k,n-1)+F(X,k-1,n-1)*X[n] which you can solve using dynamic programming. Complexity = O(kn). End conditions for F(X,k,n): When n=k F(X,k,k) = X[1]* X[2]*...*X[n]

A friend asked me a question today about an assignment problem. I found a quite straightforward solution, but I feel that it can be made simpler and faster. Your help would be appreciated. The problem: Assuming that I have N people, I need to assign them into M buildings, each building can house K people. Not all people are willing to live with each other, so i have a matrix of N*N cells and a 1 that marks the people that are willing to live with each other. If a cell contains 1 it means that I and J can live together. Obviously the matrix is symmetrical around the main diagonal. My solution

I'm working on a Free Code Camp problem - http://www.freecodecamp.com/challenges/bonfire-no-repeats-please The problem description is as follows - Return the number of total permutations of the provided string that don't have repeated consecutive letters. For example, 'aab' should return 2 because it has 6 total permutations, but only 2 of them don't have the same letter (in this case 'a') repeating. I know I can solve this by writing a program that creates every permutation and then filters out the ones with repeated characters. But I have this gnawing feeling that I can solve this

I want to know whether the task explained below is even theoretically possible, and if so how I could do it. You are given a space of N elements (i.e. all numbers between 0 and N-1 .) Let's look at the space of all permutations on that space, and call it S . The i th member of S , which can be marked S[i] , is the permutation with the lexicographic number i . For example, if N is 3, then S is this list of permutations: S[0]: 0, 1, 2 S[1]: 0, 2, 1 S[2]: 1, 0, 2 S[3]: 1, 2, 0 S[4]: 2, 0, 1 S[5]: 2, 1, 0 (Of course, when looking at a big N , this space becomes very large, N! to be exact.) Now, I

I am looking for an algorithm that can map a number to a unique permutation of a sequence. I have found out about Lehmer codes and the factorial number system thanks to a similar question, Fast permutation -> number -> permutation mapping algorithms , but that question doesn't deal with the case where there are duplicate elements in the sequence. For example, take the sequence 'AAABBC'. There are 6! = 720 ways that could be arranged, but I believe there are only 6! / (3! * 2! * 1!) = 60 unique permutation of this sequence. How can I map a number to a permutation in these cases? Edit: changed

I have a database of 20 million users and connections between those people. How can I prove the concept of "Six degrees of separation" concept in the most efficient way in programming? link to the article about Six degrees of separation You just want to measure the diameter of the graph. This is exactly the metric to find out the seperation between the most-distantly-connected nodes in a graph. Lots of algorithms on Google, Boost graph too. You can probably fit the graph in memory (in the representation that each vertex knows a list of its neighbors). Then, from each vertex n , you can run a