Features

1. Library contains generic vectors and generators. So it may generate combinatorial objects of any types. Type of the elements has to be specified as a parameter
2. Library can generate the following combinatorial objects:
   • All possible simple combinations of a vector
   • All possible multi-combinations (with repetitions) of a vector
   • All possible permutations with and without repetitions (tuples) of a vector
   • All possible sub-sets of a set or a list (with duplicates)
   • All possible compositions of an integer value
   • All possible partitions of an integer value
   • All possible partitions of a list as non-overlapping sublists
3. Library supports filtering the generated vectors
4. All generators can be used in the "for-each" loops

1. Simple permutations

This is an example of the permutations of 3 string items (apple, orange, cherry):

// Create the initial vector of 3 elements (apple, orange, cherry)
ICombinatoricsVector<String> originalVector = Factory.createVector(new String[] { "apple", "orange", "cherry" });

// Create the permutation generator by calling the appropriate method of the Factory class
Generator<String> gen = Factory.createPermutationGenerator(originalVector);

// Print the result
for (ICombinatoricsVector<String> perm : gen)
	System.out.println(perm);

And the result

  CombinatoricsVector=([apple, orange, cherry], size=3)
  CombinatoricsVector=([apple, cherry, orange], size=3)
  CombinatoricsVector=([cherry, apple, orange], size=3)
  CombinatoricsVector=([cherry, orange, apple], size=3)
  CombinatoricsVector=([orange, cherry, apple], size=3)
  CombinatoricsVector=([orange, apple, cherry], size=3)

2. Permutations with repetitions

The permutation may have more elements than slots. For example, the three possible permutation of 12 in three slots are: 111, 211, 121, 221, 112, 212, 122, and 222.

Let's generate all possible permutations with repetitions of 3 elements from the set of apple and orange:

// Create the initial vector of 2 elements (apple, orange)
ICombinatoricsVector<String> originalVector = Factory.createVector(new String[] { "apple", "orange" });

// Create the generator by calling the appropriate method in the Factory class. 
// Set the second parameter as 3, since we will generate 3-elemets permutations
Generator<String> gen = Factory.createPermutationWithRepetitionGenerator(originalVector, 3);

// Print the result
for (ICombinatoricsVector<String> perm : gen)
   System.out.println( perm );

And the result

  CombinatoricsVector=([apple, apple, apple], size=3)
  CombinatoricsVector=([orange, apple, apple], size=3)
  CombinatoricsVector=([apple, orange, apple], size=3)
  CombinatoricsVector=([orange, orange, apple], size=3)
  CombinatoricsVector=([apple, apple, orange], size=3)
  CombinatoricsVector=([orange, apple, orange], size=3)
  CombinatoricsVector=([apple, orange, orange], size=3)
  CombinatoricsVector=([orange, orange, orange], size=3)

3. Simple combinations

A simple k-combination of a finite set S is a subset of k distinct elements of S. Specifying a subset does not arrange them in a particular order. As an example, a poker hand can be described as a 5-combination of cards from a 52-card deck: the 5 cards of the hand are all distinct, and the order of the cards in the hand does not matter.

For example, let's generate 3-combination of the set (red, black, white, green, blue).

// Create the initial vector
ICombinatoricsVector<String> initialVector = Factory.createVector(new String[] {
		"red", "black", "white", "green", "blue" });

// Create a simple combination generator to generate 3-combinations of the
// initial vector
Generator<String> gen = Factory.createSimpleCombinationGenerator(initialVector, 3);

// Print all possible combinations
for (ICombinatoricsVector<String> combination : gen) {
	System.out.println(combination);
}

And the result of 10 combinations

   CombinatoricsVector=([red, black, white], size=3)
   CombinatoricsVector=([red, black, green], size=3)
   CombinatoricsVector=([red, black, blue], size=3)
   CombinatoricsVector=([red, white, green], size=3)
   CombinatoricsVector=([red, white, blue], size=3)
   CombinatoricsVector=([red, green, blue], size=3)
   CombinatoricsVector=([black, white, green], size=3)
   CombinatoricsVector=([black, white, blue], size=3)
   CombinatoricsVector=([black, green, blue], size=3)
   CombinatoricsVector=([white, green, blue], size=3)

4. Multi-combinations

As an example. Suppose there are 2 types of fruits (apple and orange) at a grocery store, and you want to buy 3 pieces of fruit. You could select

• (apple, apple, apple)
• (apple, apple, orange)
• (apple, orange, orange)
• (orange, orange, orange)

Generate 3-combinations with repetitions of the set (apple, orange).

// Create the initial vector of (apple, orange)
ICombinatoricsVector<String> initialVector = Factory.createVector(new String[] {
		"apple", "orange" });

// Create a multi-combination generator to generate 3-combinations of
// the initial vector
Generator<String> gen = Factory.createMultiCombinationGenerator(initialVector,3);
 
// Print all possible combinations
for (ICombinatoricsVector<String> combination : gen) {
	System.out.println(combination);
}

And the result of 4 multi-combinations

   CombinatoricsVector=([apple, apple, apple], size=3)
   CombinatoricsVector=([apple, apple, orange], size=3)
   CombinatoricsVector=([apple, orange, orange], size=3)
   CombinatoricsVector=([orange, orange, orange], size=3)

5. Subsets

A set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment.

Examples:

• The set (1, 2) is a proper subset of (1, 2, 3).
• Any set is a subset of itself, but not a proper subset.
• The empty set, denoted by (), is also a subset of any given set X.

All subsets of (1, 2, 3) are:

1. ()
2. (1)
3. (2)
4. (1, 2)
5. (3)
6. (1, 3)
7. (2, 3)
8. (1, 2, 3)

And code which generates all subsets of (one, two, three)

// Create an initial vector/set
ICombinatoricsVector<String> initialSet = Factory.createVector(new String[] {
		"one", "two", "three" });

// Create an instance of the subset generator
Generator<String> gen = Factory.createSubSetGenerator(initialSet);

// Print the subsets
for (ICombinatoricsVector<String> subSet : gen) {
	System.out.println(subSet);
}

And the result of all 8 possible subsets

 CombinatoricsVector=([], size=0)
 CombinatoricsVector=([one], size=1)
 CombinatoricsVector=([two], size=1)
 CombinatoricsVector=([one, two], size=2)
 CombinatoricsVector=([three], size=1)
 CombinatoricsVector=([one, three], size=2)
 CombinatoricsVector=([two, three], size=2)
 CombinatoricsVector=([one, two, three], size=3)

Version 2.0 of the combinatoricslib supports sets with duplicates. For example, if the original vector contains duplicates like (a, b, a, c), then the result will contain 14 subsets (instead of 16):

()
(a)
(b)
(a, b)
(a, a)
(b, a)
(a, b, a)
(c)
(a, c)
(b, c)
(a, b, c)
(a, a, c)
(b, a, c)
(a, b, a, c)

If you still would like to treat the set with duplicates as not identical, you should create a generator and set the second parameter of the method Factory.createSubSetGenerator() as false. In this case all 16 subsets will be generated.

Note. If the initial vector contains duplicates then the method getNumberOfGeneratedObjects won't be able to return the number of the sub sets/lists. It will throw a runtime exception

6. Integer partitions

In number theory, a partition of a positive integer n is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a composition. A summand in a partition is also called a part.

WARNING! Be careful because number of all partitions can be very high even for not great given N.

The partitions of 5 are listed below:

1. 1 + 1 + 1 + 1 + 1
2. 2 + 1 + 1 + 1
3. 2 + 2 + 1
4. 3 + 1 + 1
5. 3 + 2
6. 4 + 1
7. 5

The number of partitions of n is given by the partition function p(n). In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers.

Let's generate all possible partitions of 5:

// Create an instance of the partition generator to generate all
// possible partitions of 5
Generator<Integer> gen = Factory.createPartitionGenerator(5);

// Print the partitions
for (ICombinatoricsVector<Integer> p : gen) {
	System.out.println(p);
}

And the result of all 7 integer possible partitions

  CombinatoricsVector=([1, 1, 1, 1, 1], size=5)
  CombinatoricsVector=([2, 1, 1, 1], size=4)
  CombinatoricsVector=([2, 2, 1], size=3)
  CombinatoricsVector=([3, 1, 1], size=3)
  CombinatoricsVector=([3, 2], size=2)
  CombinatoricsVector=([4, 1], size=2)
  CombinatoricsVector=([5], size=1)

7. Integer compositions

A composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same partition of that number.

The sixteen compositions of 5 are:

1. 5
2. 4+1
3. 3+2
4. 3+1+1
5. 2+3
6. 2+2+1
7. 2+1+2
8. 2+1+1+1
9. 1+4
10. 1+3+1
11. 1+2+2
12. 1+2+1+1
13. 1+1+3
14. 1+1+2+1
15. 1+1+1+2
16. 1+1+1+1+1.

Compare this with the seven partitions of 5:

1. 5
2. 4+1
3. 3+2
4. 3+1+1
5. 2+2+1
6. 2+1+1+1
7. 1+1+1+1+1.

Example. Generate compositions all possible integer compositions of 5.

  // Create an instance of the integer composition generator to generate all possible compositions of 5
  Generator<Integer> gen = Factory.createCompositionGenerator(5);

  // Print the compositions
  for (ICombinatoricsVector<Integer> p : gen) {
     System.out.println(p);
  }

And the result

   CombinatoricsVector=([5], size=1)
   CombinatoricsVector=([1, 4], size=2)
   CombinatoricsVector=([2, 3], size=2)
   CombinatoricsVector=([1, 1, 3], size=3)
   CombinatoricsVector=([3, 2], size=2)
   CombinatoricsVector=([1, 2, 2], size=3)
   CombinatoricsVector=([2, 1, 2], size=3)
   CombinatoricsVector=([1, 1, 1, 2], size=4)
   CombinatoricsVector=([4, 1], size=2)
   CombinatoricsVector=([1, 3, 1], size=3)
   CombinatoricsVector=([2, 2, 1], size=3)
   CombinatoricsVector=([1, 1, 2, 1], size=4)
   CombinatoricsVector=([3, 1, 1], size=3)
   CombinatoricsVector=([1, 2, 1, 1], size=4)
   CombinatoricsVector=([2, 1, 1, 1], size=4)
   CombinatoricsVector=([1, 1, 1, 1, 1], size=5)

8. List Partitions

It is possible to generate non-overlapping sublists of length n of a given list

For example, if a list is (A, B, B, C), then the non-overlapping sublists of length 2 will be:

1. ( (A), (B, B, C) )
2. ( (B, B, C), (A) )
3. ( (B), (A, B, C) )
4. ( (A, B, C), (B) )
5. ( (A, B), (B, C) )
6. ( (B, C), (A, B) )
7. ( (B, B), (A, C) )
8. ( (A, C), (B, B) )
9. ( (A, B, B), (C) )
10. ( (C), (A, B, B) )

To do that you should use an instance of the complex combination generator

  // create a vector (A, B, B, C)
  ICombinatoricsVector<String> vector = Factory.createVector(new String[] { "A", "B", "B", "C" });

  // Create a complex-combination generator
  Generator<ICombinatoricsVector<String>> gen = new ComplexCombinationGenerator<String>(vector, 2);

  // Iterate the combinations
  for (ICombinatoricsVector<ICombinatoricsVector<String>> comb : gen) {
     System.out.println(ComplexCombinationGenerator.convert2String(comb) + " - " + comb);
  }

And the result

 ([A],[B, B, C]) - CombinatoricsVector=([CombinatoricsVector=([A], size=1), CombinatoricsVector=([B, B, C], size=3)], size=2)
 ([B, B, C],[A]) - CombinatoricsVector=([CombinatoricsVector=([B, B, C], size=3), CombinatoricsVector=([A], size=1)], size=2)
 ([B],[A, B, C]) - CombinatoricsVector=([CombinatoricsVector=([B], size=1), CombinatoricsVector=([A, B, C], size=3)], size=2)
 ([A, B, C],[B]) - CombinatoricsVector=([CombinatoricsVector=([A, B, C], size=3), CombinatoricsVector=([B], size=1)], size=2)
 ([A, B],[B, C]) - CombinatoricsVector=([CombinatoricsVector=([A, B], size=2), CombinatoricsVector=([B, C], size=2)], size=2)
 ([B, C],[A, B]) - CombinatoricsVector=([CombinatoricsVector=([B, C], size=2), CombinatoricsVector=([A, B], size=2)], size=2)
 ([B, B],[A, C]) - CombinatoricsVector=([CombinatoricsVector=([B, B], size=2), CombinatoricsVector=([A, C], size=2)], size=2)
 ([A, C],[B, B]) - CombinatoricsVector=([CombinatoricsVector=([A, C], size=2), CombinatoricsVector=([B, B], size=2)], size=2)
 ([A, B, B],[C]) - CombinatoricsVector=([CombinatoricsVector=([A, B, B], size=3), CombinatoricsVector=([C], size=1)], size=2)
 ([C],[A, B, B]) - CombinatoricsVector=([CombinatoricsVector=([C], size=1), CombinatoricsVector=([A, B, B], size=3)], size=2)