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Rank of a Permutation

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.

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Rank of a Permutation -

Tag(s) : Combinatorics

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Rank of a Permutation

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See also: Permutations

Find a Permutation with its Rank/Order



See also: Permutations

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.

Answers to Questions

What is the rank of a permutation?

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:

0ABC
1ACB
2BAC
3BCA
4CAB
5CBA
Thus, the permutation BAC is at rank number 2 (starting at 0)

How to calculate the rank of a permutation?

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

Take 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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