Algorithm to determine possible groups of items

Question

I am scratching my head trying to do this and it\'s eating me up. I know it is not THAT complex. I have a number of items, this number can be equal or greater than three. Then I

Answer 1

Here's an algorithm (expressed in C++) to solve a more general version of the problem, with arbitrary upper and lower bounds on the addends that may appear in each partition:

#include 
#include 

using namespace std;

typedef vector Partition;
typedef vector Partition_list;

// Count and return all partitions of an integer N using only 
// addends between min and max inclusive.

int p(int min, int max, int n, Partition_list &v)
{
   if (min > max) return 0;
   if (min > n) return 0;     
   if (min == n) {
      Partition vtemp(1,min);
      v.push_back(vtemp);
      return 1;
   }
   else {
     Partition_list part1,part2;
     int p1 = p(min+1,max,n,part1);
     int p2 = p(min,max,n-min,part2);
     v.insert(v.end(),part1.begin(),part1.end());
     for(int i=0; i < p2; i++)
     {
        part2[i].push_back(min);
     }
     v.insert(v.end(),part2.begin(),part2.end());
     return p1+p2;
   }
}

void print_partition(Partition &p)
{
   for(int i=0; i < p.size(); i++) {
      cout << p[i] << ' ';
   }
   cout << "\n";
}

void print_partition_list(Partition_list &pl)
{
   for(int i = 0; i < pl.size(); i++) {
      print_partition(pl[i]);
   }
}

int main(int argc, char **argv)
{
   Partition_list v_master;

   int n = atoi(argv[1]);
   int min = atoi(argv[2]);
   int max = atoi(argv[3]);
   int count = p(min,max,n,v_master);
   cout << count << " partitions of " << n << " with min " << min  ;
   cout << " and max " << max << ":\n" ;
   print_partition_list(v_master);
}

And some sample output:

$ ./partitions 12 3 7              
6 partitions of 12 with min 3 and max 7:
6 6 
7 5 
4 4 4 
5 4 3 
6 3 3 
3 3 3 3 
$ ./partitions 50 10 20            
38 partitions of 50 with min 10 and max 20:
17 17 16 
18 16 16 
18 17 15 
19 16 15 
20 15 15 
18 18 14 
19 17 14 
20 16 14 
19 18 13 
20 17 13 
19 19 12 
20 18 12 
13 13 12 12 
14 12 12 12 
20 19 11 
13 13 13 11 
14 13 12 11 
15 12 12 11 
14 14 11 11 
15 13 11 11 
16 12 11 11 
17 11 11 11 
20 20 10 
14 13 13 10 
14 14 12 10 
15 13 12 10 
16 12 12 10 
15 14 11 10 
16 13 11 10 
17 12 11 10 
18 11 11 10 
15 15 10 10 
16 14 10 10 
17 13 10 10 
18 12 10 10 
19 11 10 10 
20 10 10 10 
10 10 10 10 10