CF715E Complete the Permutations

ZS the Coder is given two permutations $ p $ and $ q $ of $ {1,2,...,n} $ , but some of their elements are replaced with $ 0 $ . The distance between two permutations $ p $ and $ q $ is defined as the minimum number of moves required to turn $ p $ into $ q $ . A move consists of swapping exactly $ 2 $ elements of $ p $ . ZS the Coder wants to determine the number of ways to replace the zeros with positive integers from the set $ {1,2,...,n} $ such that $ p $ and $ q $ are permutations of $ {1,2,...,n} $ and the distance between $ p $ and $ q $ is exactly $ k $ . ZS the Coder wants to find the answer for all $ 0

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In the first sample case, there is the only way to replace zeros so that it takes $ 0 $ swaps to convert $ p $ into $ q $ , namely $ p=(1,2,3),q=(1,2,3) $ . There are two ways to replace zeros so that it takes $ 1 $ swap to turn $ p $ into $ q $ . One of these ways is $ p=(1,2,3),q=(3,2,1) $ , then swapping $ 1 $ and $ 3 $ from $ p $ transform it into $ q $ . The other way is $ p=(1,3,2),q=(1,2,3) $ . Swapping $ 2 $ and $ 3 $ works in this case. Finally, there is one way to replace zeros so that it takes $ 2 $ swaps to turn $ p $ into $ q $ , namely $ p=(1,3,2),q=(3,2,1) $ . Then, we can transform $ p $ into $ q $ like following: .