CF1827F Copium Permutation

You are given a permutation $ a_1,a_2,\ldots,a_n $ of the first $ n $ positive integers. A subarray $ [l,r] $ is called copium if we can rearrange it so that it becomes a sequence of consecutive integers, or more formally, if $ $$$\max(a_l,a_{l+1},\ldots,a_r)-\min(a_l,a_{l+1},\ldots,a_r)=r-l $ $ For each $ k $ in the range $ \[0,n\] $ , print out the maximum number of copium subarrays of $ a $ over all ways of rearranging the last $ n-k $ elements of $ a$$$.

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^4 $ ). The description of the test cases follows. The first line of each test case contains a single integer $ n $ ( $ 1\le n\le 2\cdot 10^5 $ ). The second line of each test case contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 1 \le a_i \le n $ ). It is guaranteed that the given numbers form a permutation of length $ n $ . It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2\cdot 10^5 $ .

For each test case print $ n+1 $ integers as the answers for each $ k $ in the range $ [0,n] $ .

In the first test case, the answer permutations for each $ k $ are $ [1,2,3,4,5] $ , $ [5,4,3,2,1] $ , $ [5,2,3,4,1] $ , $ [5,2,1,3,4] $ , $ [5,2,1,4,3] $ , $ [5,2,1,4,3] $ . In the second test case, the answer permutations for each $ k $ are $ [1,2,3,4] $ , $ [2,1,3,4] $ , $ [2,1,3,4] $ , $ [2,1,4,3] $ , $ [2,1,4,3] $ .