A13374 | Constructing the Array

You are given an array $a$ of length $n$ consisting of zeros. You perform $n$ actions with this array: during the $i$ -th action, the following sequence of operations appears:

1. Choose the maximum by length subarray (continuous subsegment) consisting only of zeros, among all such segments choose the leftmost one;
2. Let this segment be $[l; r]$ . If $r-l+1$ is odd (not divisible by $2$ ) then assign (set) $a[\frac{l+r}{2}] := i$ (where $i$ is the number of the current action), otherwise (if $r-l+1$ is even) assign (set) $a[\frac{l+r-1}{2}] := i$ .

Consider the array $a$ of length $5$ (initially $a=[0, 0, 0, 0, 0]$ ). Then it changes as follows:

1. Firstly, we choose the segment $[1; 5]$ and assign $a[3] := 1$ , so $a$ becomes $[0, 0, 1, 0, 0]$ ;
2. then we choose the segment $[1; 2]$ and assign $a[1] := 2$ , so $a$ becomes $[2, 0, 1, 0, 0]$ ;
3. then we choose the segment $[4; 5]$ and assign $a[4] := 3$ , so $a$ becomes $[2, 0, 1, 3, 0]$ ;
4. then we choose the segment $[2; 2]$ and assign $a[2] := 4$ , so $a$ becomes $[2, 4, 1, 3, 0]$ ;
5. and at last we choose the segment $[5; 5]$ and assign $a[5] := 5$ , so $a$ becomes $[2, 4, 1, 3, 5]$ .

Your task is to find the array $a$ of length $n$ after performing all $n$ actions. Note that the answer exists and unique.

You have to answer $t$ independent test cases.

The first line of the input contains one integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases. Then $t$ test cases follow.

The only line of the test case contains one integer $n$ ( $1 \le n \le 2 \cdot 10^5$ ) — the length of $a$ .

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ ( $\sum n \le 2 \cdot 10^5$ ).

For each test case, print the answer — the array $a$ of length $n$ after performing $n$ actions described in the problem statement. Note that the answer exists and unique.