Number of n-element permutations with exactly k inversions

Question

I am trying to efficiently solve SPOJ Problem 64: Permutations.

Let A = [a1,a2,...,an] be a permutation of integers 1,2,...,n. A pair of indices (i,

Answer 1

If there is a dynamic programming solution, there is probably a way to do it step by step, using the results for permutations of length n to help with the results for permutations of length n+1.

Given a permutation of length n - values 1-n, you can get a permutation of length n+1 by adding value (n+1) at n+1 possible positions. (n+1) is larger than any of 1-n so the number of inversions you create when you do this depends on where you add it - add it at the last position and you create no inversions, add it at the last but one position and you create one inversion, and so on - look back at the n=4 cases with one inversion to check this.

So if you consider one of n+1 places where you can add (n+1) if you add it at place j counting from the right so the last position as position 0 the number of permutations with K inversions this creates is the number of permutations with K-j inversions on n places.

So if at each step you count the number of permutations with K inversions for all possible K you can update the number of permutations with K inversions for length n+1 using the number of permutations with K inversions for length n.