A15343 | Permutation Chain

A permutation of length $n$ is a sequence of integers from $1$ to $n$ such that each integer appears in it exactly once.

Let the fixedness of a permutation $p$ be the number of fixed points in it — the number of positions $j$ such that $p_j = j$ , where $p_j$ is the $j$ -th element of the permutation $p$ .

You are asked to build a sequence of permutations $a_1, a_2, \dots$ , starting from the identity permutation (permutation $a_1 = [1, 2, \dots, n]$ ). Let's call it a permutation chain. Thus, $a_i$ is the $i$ -th permutation of length $n$ .

For every $i$ from $2$ onwards, the permutation $a_i$ should be obtained from the permutation $a_{i-1}$ by swapping any two elements in it (not necessarily neighboring). The fixedness of the permutation $a_i$ should be strictly lower than the fixedness of the permutation $a_{i-1}$ .

Consider some chains for $n = 3$ :

- $a_1 = [1, 2, 3]$ , $a_2 = [1, 3, 2]$ — that is a valid chain of length $2$ . From $a_1$ to $a_2$ , the elements on positions $2$ and $3$ get swapped, the fixedness decrease from $3$ to $1$ .
- $a_1 = [2, 1, 3]$ , $a_2 = [3, 1, 2]$ — that is not a valid chain. The first permutation should always be $[1, 2, 3]$ for $n = 3$ .
- $a_1 = [1, 2, 3]$ , $a_2 = [1, 3, 2]$ , $a_3 = [1, 2, 3]$ — that is not a valid chain. From $a_2$ to $a_3$ , the elements on positions $2$ and $3$ get swapped but the fixedness increase from $1$ to $3$ .
- $a_1 = [1, 2, 3]$ , $a_2 = [3, 2, 1]$ , $a_3 = [3, 1, 2]$ — that is a valid chain of length $3$ . From $a_1$ to $a_2$ , the elements on positions $1$ and $3$ get swapped, the fixedness decrease from $3$ to $1$ . From $a_2$ to $a_3$ , the elements on positions $2$ and $3$ get swapped, the fixedness decrease from $1$ to $0$ .

Find the longest permutation chain. If there are multiple longest answers, print any of them.

The first line contains a single integer $t$ ( $1 \le t \le 99$ ) — the number of testcases.

The only line of each testcase contains a single integer $n$ ( $2 \le n \le 100$ ) — the required length of permutations in the chain.

For each testcase, first, print the length of a permutation chain $k$ .

Then print $k$ permutations $a_1, a_2, \dots, a_k$ . $a_1$ should be an identity permutation of length $n$ ( $[1, 2, \dots, n]$ ). For each $i$ from $2$ to $k$ , $a_i$ should be obtained by swapping two elements in $a_{i-1}$ . It should also have a strictly lower fixedness than $a_{i-1}$ .