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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) : Mathematics, Combinatorics

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# Rank of a Permutation

## Find a Permutation with its Rank

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.

### How to calculate the rank of a permutation?

To find the row of a permutation, list all possible permutations and sort them in ascending order.

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

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

Example: The permutation BAC is at number 2 (starting at 0)

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 $$s_B + s_A + s_C = 2 + 0 + 0 = 2$$