CF2123E MEX Count

Define the $ \mathrm{MEX} $ (minimum excluded value) of an array to be the smallest nonnegative integer not present in the array. For example, - $ \mathrm{MEX}([2, 2, 1]) = 0 $ because $ 0 $ is not in the array. - $ \mathrm{MEX}([3, 1, 0, 1]) = 2 $ because $ 0 $ and $ 1 $ are in the array but $ 2 $ is not. - $ \mathrm{MEX}([0, 3, 1, 2]) = 4 $ because $ 0 $ , $ 1 $ , $ 2 $ , and $ 3 $ are in the array but $ 4 $ is not. You are given an array $ a $ of size $ n $ of nonnegative integers. For all $ k $ ( $ 0\leq k \leq n $ ), count the number of possible values of $ \mathrm{MEX}(a) $ after removing exactly $ k $ values from $ a $ .

The first line contains an integer $ t $ ( $ 1\leq t\leq 10^4 $ ) — the number of test cases. The first line of each test case contains one integer $ n $ ( $ 1\leq n\leq 2\cdot 10^5 $ ) — the size of the array $ a $ . The second line of each test case contains $ n $ integers, $ a_1,a_2,\dots,a_n $ ( $ 0\leq a_i\leq n $ ). It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2\cdot 10^5 $ .

For each test case, output a single line containing $ n+1 $ integers — the number of possible values of $ \mathrm{MEX}(a) $ after removing exactly $ k $ values, for $ k=0,1,\dots,n $ .

In the first sample, consider $ k=1 $ . If you remove a $ 0 $ , then you get the following array: 1012 So we get $ \mathrm{MEX}(a) = 3 $ . Alternatively, if you remove the $ 2 $ , then you get the following array: 1001 So we get $ \mathrm{MEX}(a) = 2 $ . It can be shown that these are the only possible values of $ \mathrm{MEX}(a) $ after removing exactly one value. So the output for $ k=1 $ is $ 2 $ .