Roundgod has a permutation of length . She plans to sort the permutation in ascending order with an algorithm depends on luck. At any time, Roundgod could spend  money to choose a subset  of indices and then uniformly shuffle the values in corresponding positions, where  means the size of . She wants to know the expected value of minimum cost to sort the permutation under the optimal decision. 

Only one permutation can't show Roundgod's luck clearly, so lzr gives Roundgod  permutations. It can be proved that the expected values can always be written in reduced fraction , where  is not a multiple of . Please tell Roundgod the answer for each permutation by  , where is the multiplicative inverse modulo .