combinatorics

I need to generate all the partitions of a given integer. I found this algorithm by Jerome Kelleher for which it is stated to be the most efficient one: def accelAsc(n): a = [0 for i in range(n + 1)] k = 1 a[0] = 0 y = n - 1 while k != 0: x = a[k - 1] + 1 k -= 1 while 2*x <= y: a[k] = x y -= x k += 1 l = k + 1 while x <= y: a[k] = x a[l] = y yield a[:k + 2] x += 1 y -= 1 a[k] = x + y y = x + y - 1 yield a[:k + 1] reference: http://homepages.ed.ac.uk/jkellehe/partitions.php By the way, it is not quite efficient. For an input like 40 it freezes nearly my whole system for few seconds before

I have pools of values and I would like to generate every possible unordered combination by picking from certain pools. For example, I wanted to pick from pool 0, pool 0, and pool 1: >>> pools = [[1, 2, 3], [2, 3, 4], [3, 4, 5]] >>> part = (0, 0, 1) >>> list(product(*(pools[i] for i in part))) [(1, 1, 2), (1, 1, 3), (1, 1, 4), (1, 2, 2), (1, 2, 3), (1, 2, 4), (1, 3, 2), (1, 3, 3), (1, 3, 4), (2, 1, 2), (2, 1, 3), (2, 1, 4), (2, 2, 2), (2, 2, 3), (2, 2, 4), (2, 3, 2), (2, 3, 3), (2, 3, 4), (3, 1, 2), (3, 1, 3), (3, 1, 4), (3, 2, 2), (3, 2, 3), (3, 2, 4), (3, 3, 2), (3, 3, 3), (3, 3, 4)] This

Suppose we have a table of numbers like this (we can assume it is a square table): 20 2 1 3 4 5 1 14 8 9 15 12 17 17 11 16 1 1 15 18 20 13 15 5 11 Your job is to calculate the maximum sum of n numbers where n is the number of rows or columns in the table. The catch is each number must come from a unique row and column. For example, selecting the numbers at (0,0), (1,1), (2,2), (3,3), and (4,4) is acceptable, but (0,0), (0,1), (2,2), (3,3), and (4,4) is not because the first two numbers were pulled from the same row. My (laughable) solution to this problem is iterating through all the possible

I was recently faced with a prompt for a programming algorithm that I had no idea what to do for. I've never really written an algorithm before, so I'm kind of a newb at this. The problem said to write a program to determine all of the possible coin combinations for a cashier to give back as change based on coin values and number of coins. For example, there could be a currency with 4 coins: a 2 cent, 6 cent, 10 cent and 15 cent coins. How many combinations of this that equal 50 cents are there? The language I'm using is C++, although that doesn't really matter too much. edit: This is a more

I am trying to efficiently solve SPOJ Problem 64: Permutations . Let A = [a1,a2,...,an] be a permutation of integers 1,2,...,n. A pair of indices (i,j), 1<=i<=j<=n, is an inversion of the permutation A if ai>aj. We are given integers n>0 and k>=0. What is the number of n-element permutations containing exactly k inversions? For instance, the number of 4-element permutations with exactly 1 inversion equals 3. To make the given example easier to see, here are the three 4-element permutations with exactly 1 inversion: (1, 2, 4, 3) (1, 3, 2, 4) (2, 1, 3, 4) In the first permutation, 4 > 3 and the