CF1279E New Year Permutations

Yeah, we failed to make up a New Year legend for this problem. A permutation of length $ n $ is an array of $ n $ integers such that every integer from $ 1 $ to $ n $ appears in it exactly once. An element $ y $ of permutation $ p $ is reachable from element $ x $ if $ x = y $ , or $ p_x = y $ , or $ p_{p_x} = y $ , and so on. The decomposition of a permutation $ p $ is defined as follows: firstly, we have a permutation $ p $ , all elements of which are not marked, and an empty list $ l $ . Then we do the following: while there is at least one not marked element in $ p $ , we find the leftmost such element, list all elements that are reachable from it in the order they appear in $ p $ , mark all of these elements, then cyclically shift the list of those elements so that the maximum appears at the first position, and add this list as an element of $ l $ . After all elements are marked, $ l $ is the result of this decomposition. For example, if we want to build a decomposition of $ p = [5, 4, 2, 3, 1, 7, 8, 6] $ , we do the following: 1. initially $ p = [5, 4, 2, 3, 1, 7, 8, 6] $ (bold elements are marked), $ l = [] $ ; 2. the leftmost unmarked element is $ 5 $ ; $ 5 $ and $ 1 $ are reachable from it, so the list we want to shift is $ [5, 1] $ ; there is no need to shift it, since maximum is already the first element; 3. $ p = [\textbf{5}, 4, 2, 3, \textbf{1}, 7, 8, 6] $ , $ l = [[5, 1]] $ ; 4. the leftmost unmarked element is $ 4 $ , the list of reachable elements is $ [4, 2, 3] $ ; the maximum is already the first element, so there's no need to shift it; 5. $ p = [\textbf{5}, \textbf{4}, \textbf{2}, \textbf{3}, \textbf{1}, 7, 8, 6] $ , $ l = [[5, 1], [4, 2, 3]] $ ; 6. the leftmost unmarked element is $ 7 $ , the list of reachable elements is $ [7, 8, 6] $ ; we have to shift it, so it becomes $ [8, 6, 7] $ ; 7. $ p = [\textbf{5}, \textbf{4}, \textbf{2}, \textbf{3}, \textbf{1}, \textbf{7}, \textbf{8}, \textbf{6}] $ , $ l = [[5, 1], [4, 2, 3], [8, 6, 7]] $ ; 8. all elements are marked, so $ [[5, 1], [4, 2, 3], [8, 6, 7]] $ is the result. The New Year transformation of a permutation is defined as follows: we build the decomposition of this permutation; then we sort all lists in decomposition in ascending order of the first elements (we don't swap the elements in these lists, only the lists themselves); then we concatenate the lists into one list which becomes a new permutation. For example, the New Year transformation of $ p = [5, 4, 2, 3, 1, 7, 8, 6] $ is built as follows: 1. the decomposition is $ [[5, 1], [4, 2, 3], [8, 6, 7]] $ ; 2. after sorting the decomposition, it becomes $ [[4, 2, 3], [5, 1], [8, 6, 7]] $ ; 3. $ [4, 2, 3, 5, 1, 8, 6, 7] $ is the result of the transformation. We call a permutation good if the result of its transformation is the same as the permutation itself. For example, $ [4, 3, 1, 2, 8, 5, 6, 7] $ is a good permutation; and $ [5, 4, 2, 3, 1, 7, 8, 6] $ is bad, since the result of transformation is $ [4, 2, 3, 5, 1, 8, 6, 7] $ . Your task is the following: given $ n $ and $ k $ , find the $ k $ -th (lexicographically) good permutation of length $ n $ .

The first line contains one integer $ t $ ( $ 1 \le t \le 1000 $ ) — the number of test cases. Then the test cases follow. Each test case is represented by one line containing two integers $ n $ and $ k $ ( $ 1 \le n \le 50 $ , $ 1 \le k \le 10^{18} $ ).

For each test case, print the answer to it as follows: if the number of good permutations of length $ n $ is less than $ k $ , print one integer $ -1 $ ; otherwise, print the $ k $ -th good permutation on $ n $ elements (in lexicographical order).