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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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Tag(s) : Combinatorics

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# Rank of a Permutation

## Find a Permutation with its Rank/Order

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.

### 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:

 0 ABC 1 ACB 2 BAC 3 BCA 4 CAB 5 CBA
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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