CF1516E Baby Ehab Plays with Permutations

This time around, Baby Ehab will play with permutations. He has $ n $ cubes arranged in a row, with numbers from $ 1 $ to $ n $ written on them. He'll make exactly $ j $ operations. In each operation, he'll pick up $ 2 $ cubes and switch their positions. He's wondering: how many different sequences of cubes can I have at the end? Since Baby Ehab is a turbulent person, he doesn't know how many operations he'll make, so he wants the answer for every possible $ j $ between $ 1 $ and $ k $ .

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In the second example, there are $ 3 $ sequences he can get after $ 1 $ swap, because there are $ 3 $ pairs of cubes he can swap. Also, there are $ 3 $ sequences he can get after $ 2 $ swaps: - $ [1,2,3] $ , - $ [3,1,2] $ , - $ [2,3,1] $ .