#### PROB8: Permutation Inversions

Count the number of permutations that have a specific number of inversions.

#### DESCRIPTION

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: prob8.dat

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.

An example input file would be

```       column   1
1234567890
line 1:3 0[EOL]
2:3 1[EOL]
3:3 2[EOL]
4:3 3[EOL]
5:4 2[EOL]
6:4 10[EOL]
7:13 23[EOL]
8:18 80[EOL]
9:0 0[EOL]
:[EOF]
```

#### OUTPUT: prob8.out

For each problem, output the number of permutations of {1, ..., n}with exactly k inversions.

The correct output corresponding to the example input file would be

```       column   111111111122222222223
123456789012345678901234567890
line 1:Program 8 by team 0[EOL]
2:1[EOL]
3:2[EOL]
4:2[EOL]
5:1[EOL]
6:5[EOL]
7:0[EOL]
8:46936280[EOL]
9:184348859235088[EOL]
10:End of program 8 by team 0[EOL]
:[EOF]
```

#### NOTES

Even though only integer arithmetic is performed, use double precision values to represent quantities to avoid overflows.