A13096 | 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).