Description
Count the number of permutations that have a specific number of inversions.
Given a permutation a1, a2, a3,..., an of the n integers 1, 2, 3, ..., n, an inversion is a pair (ai, aj) where i < j and ai > aj. The number of inversions in a permutation gives an indication on how "unsorted" a permutation is. If we wish to analyze the average running time of a sorting algorithm, it is often useful to know how many permutations of n objects will have a certain number of inversions.
In this problem you are asked to compute the number of permutations of n values that have exactly k inversions.
For example, if n = 3, there are 6 permutations with the indicated inversions as follows:
123 0 inversions
132 1 inversion (3 > 2)
213 1 inversion (2 > 1)
231 2 inversions (2 > 1, 3 > 1)
312 2 inversions (3 > 1, 3 > 2)
321 3 inversions (3 > 2, 3 > 1, 2 > 1)
Therefore, for the permutations of 3 things
1 of them has 0 inversions
2 of them have 1 inversion
2 of them have 2 inversions
1 of them has 3 inversions
0 of them have 4 inversions
0 of them have 5 inversions
etc.
Input
The input consists one or more problems. The input for each problem is specified on a single line, giving the integer n (1 <= n <= 18) and a non-negative integer k (0 <= k <= 200). The end of input is specified by a line with n = k = 0.
Output
For each problem, output the number of permutations of {1, ..., n}with exactly k inversions.
Sample Input
3 0
3 1
3 2
3 3
4 2
4 10
13 23
18 80
0 0
Sample Output
1
2
2
1
5
0
46936280
184348859235088
