시간 제한 | 메모리 제한 | 제출 | 정답 | 맞힌 사람 | 정답 비율 |
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1 초 | 128 MB | 33 | 25 | 22 | 75.862% |
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
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.
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