Question

I’m struggling to find efficient algorithm for calculating the rank of a permutation, and vice versa (permutation for a given rank). Can someone give some pointers?

Answer 1

Do you have repeating elements in the array?

If there are only unique elements, the following recursion calculates the rank of X[m:n] as a permutation of length n-m+1:

Rank(X, m:n) = RankOfElement(X[m], X[m:n]) * Factorial(n – m) + Rank(X, (m+1):n)

Both Rank and RankOfElement are zero-based (start at 0).

Basically, Rank(permutation) = Rank(first permutation starting with first letter of permutation) + Rank(permutation with first letter deleted), e.g. for string EDCBA that means Rank(EDCBA) = Rank(EABCD) + Rank(DCBA).

This can be extended to non-unique cases by changing the first term:

Rank(X, m:n) = Rank(X, (m+1):n) + ∑ over (y ∈ X[m:n], y < X[m]) of number of combinations of {X[m:n]}-{y} .