Rank of a Permutation

The rank of a permutation is an integer that represents the position of a specific permutation in the lexicographic order of all possible permutations of a set of distinct elements.

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.

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 $.

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.

A source code that calculates the rank of an alphabetical permutation would be:// Pseudo-code
function rankPermutation(p) {
alphabet = "abcdefghijklmnopqrstuvwxyz"
length = length(p)
rank = 0
j = 0
for i = length-1 down to 0 {
letter = p[j++]
index = position(letter, alphabet)
alphabet = alphabet[0..index] + alphabet[index+1..]
rank += index * factorial(i);
}
return rank;
}