Tool to calculate the rank of a permutation of a set. The permutation's rank is the number associated with it in the order of generation of the permutations.

Rank of a Permutation - dCode

Tag(s) : Combinatorics

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Tool to calculate the rank of a permutation of a set. The permutation's rank is the number associated with it in the order of generation of the permutations.

From the list of all possible permutations of a set (or arrangements), it is possible to sort this index in ascending order. The rank of a permutation is the position of that if in the sorted list.

Example: The set A,B,C has for permutations:

0 | ABC |

1 | ACB |

2 | BAC |

3 | BCA |

4 | CAB |

5 | CBA |

Since it seems difficult to list all permutations when there are many items. There is a mathematical method to perform this calculation.

Consider a permutation \( P \) in the set \( E \) of size \( t \).

Example: The permutation B,A,C from the initial set A,B,C of size \( t = 3 \)

For each letter, calculate the position \( p \) in the set \( E \), calculate \( s = p \times (t-1)! \) and remove the letter from the set \( E \) (size \( t \) decreases). The sum of \( s \) is the rank of the permutation.

Example: B is in position \( 1 \) in ABC, \( s_B = 1 \times 2! = 2 \)

A is in position \( 0 \) in AC, \( s_A = 0 \times 1! = 0 \)

C is in position \( 0 \) in C, \( s_C = 0 \times 0! = 0 \)

BAC is at permutation rank \( s_B + s_A + s_C = 2 + 0 + 0 = 2 \)

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Source : https://www.dcode.fr/permutation-rank

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