C＃中的部分组合/置换-例如A，B，C = {A，B}，{A，C}，{B，C}

Question

There are many brilliant implementations for Permutations - I chose Sam's answer in the link.

I also understand there is a difference between permutations and combinations but I don't know how to word this properly.

I need guidance on getting all unique partial combinations please, eg

A,B,C = {A,B}, {A,C}, {B,C}
A,B,C,D = {A,B,C},{A,B,D},{B,C,D},{A,C,D},{A,B}, {A,C}, {B,C}

From here I will pass this to the permutation function to get me all available permutations, eg {A,B}, {B,A}, {A,C}, {C,A} etc.

How can I get these (partial) subsets of the greater set?

Answer 1

It is quite easy to do recursively. You go through a virtual tree in the GetSubCombinations function, always returning the set without the current element first and then with the current element. On the last level (the first part of the GetSubCombinations function) you generate the lists that are being returned, either including the last element or being empty.

Code follows:

using System.Collections.Generic;


class Program {
    private static IEnumerable<List<char>> GetSubCombinations(char[] elements, uint startingPos)
    {
        // Leaf condition
        if (startingPos == elements.Length - 1)
        {
            yield return new List<char> {elements[startingPos]};
            yield return new List<char>();
            yield break;
        }

        // node splitting
        foreach (var list in GetSubCombinations(elements, startingPos + 1))
        {
            yield return list;
            list.Add(elements[startingPos]);
            yield return list;
            list.Remove(elements[startingPos]);
        }
    }

    private static IEnumerable<List<char>> GetPartialCombinations(char[] elements)
    {
        foreach (var c in GetSubCombinations(elements, 0))
        {
            // Here you can filter out trivial combinations,
            // e.g. all elements, individual elements and the empty set
            if (c.Count > 1 && c.Count < elements.Length)
                yield return c;
        }
    }


    static void Main(string[] args) {
        char[] elements = new char[] {'A', 'B', 'C'};
        foreach (var combination in GetPartialCombinations(elements))
        {
            foreach (char elem in combination)
                System.Console.Write(elem + ", ");
            System.Console.Write("\n");
        }
        return;
    }

}

Answer 2

The difference between permutation to a combination is that in a permutation the order matters.

From what I understand you want all the Combinations , Thus all combinations that don't fill the exact set (You want to make a subset)

EX:

So for S = {A,B,C,D} , an example of a partial combination would be {A,C,B} as it doesn't care about the order and it doesn't contain the full set (ie. it is a subset of S)

The formula for Combination is N choose K

So your set has 4 elements (N) and you want a set of 3 or less (K)

So N!/K!(N-K)! 
3: = 4!/3!(4-3)! = (1x2x3x4)/(1x2x3)(1) = 4
2: = 4!/2!(2)! = 1x2x3x4/(1x2x1x2) = 6
1: = 4!/1!(4-1)! = 1x2x3x4/1x2x3 = 4

Thus answer is 14

If you want some code on how to implement a combination calculation: SOURCE

private static void GetCombination(List<int> list)
{
    double count = Math.Pow(2, list.Count);
    for (int i = 1; i <= count - 1; i++)
    {
        string str = Convert.ToString(i, 2).PadLeft(list.Count, '0');
        for (int j = 0; j < str.Length; j++)
        {
            if (str[j] == '1')
            {
                Console.Write(list[j]);
            }
        }
        Console.WriteLine();
    }
}