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.

3 0
3 1
3 2
3 3
4 2
4 10
13 23
18 80
0 0

1
2
2
1
5
0
46936280
184348859235088

Source

Rocky Mountain 2003