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K-Permutations

Tool to count and generate partial k-permutations. In Mathematics, a partial permutation (or k-permutation) is an ordered list of items. It consists in permutations of each combination.

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K-Permutations -

Tag(s) : Combinatorics, Mathematics

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# K-Permutations

## Counting Partial Permutations

Tool to count and generate partial k-permutations. In Mathematics, a partial permutation (or k-permutation) is an ordered list of items. It consists in permutations of each combination.

### How to generate k-permutations?

A partial permutation of k items out of n consist in the list of all possible permutation of each combination of k out of n.

Example: 2 items among 3 (A,B,C) can be shuffled in 6 ways: A,B A,C B,A B,C C,A C,B

In a k-permuation the notion of order is important A,B is different from B,A, contrary to combinations.

### How to count k-permutations?

The counting of the permutations uses combinatorics and factorials

Example: For k items out of n, the number of k-permutations is equal to $$n!/(n-k)!$$

### Chat is the list of arrangements?

The list is infinite, here are some examples:

 A(2,3) 6 arranged subsets (1,2)(2,1)(1,3)(3,1)(2,3)(3,2) A(2,4) 12 arranged subsets (1,2)(2,1)(1,3)(3,1)(1,4)(4,1)(2,3)(3,2)(2,4)(4,2)(3,4)(4,3) A(2,5) 20 arranged subsets (1,2)(2,1)(1,3)(3,1)(1,4)(4,1)(1,5)(5,1)(2,3)(3,2)(2,4)(4,2)(2,5)(5,2)(3,4)(4,3)(3,5)(5,3)(4,5)(5,4) A(2,6) 30 arranged subsets (1,2)(2,1)(1,3)(3,1)(1,4)(4,1)(1,5)(5,1)(1,6)(6,1)(2,3)(3,2)(2,4)(4,2)(2,5)(5,2)(2,6)(6,2)(3,4)(4,3)(3,5)(5,3)(3,6)(6,3)(4,5)(5,4)(4,6)(6,4)(5,6)(6,5) A(3,4) 24 arranged subsets (1,2,3)(2,1,3)(3,1,2)(1,3,2)(2,3,1)(3,2,1)(1,2,4)(2,1,4)(4,1,2)(1,4,2)(2,4,1)(4,2,1)(1,3,4)(3,1,4)(4,1,3)(1,4,3)(3,4,1)(4,3,1)(2,3,4)(3,2,4)(4,2,3)(2,4,3)(3,4,2)(4,3,2)

### How to remove the limit when computing k-permutations?

Partial permutations need exponential resources of computer time, so the generation must be paid.