## k-Permutation Generator

## Counting Partial Permutations

## Answers to Questions

### 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.

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